id sid tid token lemma pos ejpam-308 1 1 2_308_sury.dvi 2_308_sury.dvi NUM ejpam-308 1 2 european european ADJ ejpam-308 1 3 journal journal NOUN ejpam-308 1 4 of of ADP ejpam-308 1 5 pure pure ADJ ejpam-308 1 6 and and CCONJ ejpam-308 1 7 applied apply VERB ejpam-308 1 8 mathematics mathematic NOUN ejpam-308 1 9 vol vol NOUN ejpam-308 1 10 . . PUNCT ejpam-308 2 1 3 3 NUM ejpam-308 2 2 , , PUNCT ejpam-308 2 3 no no INTJ ejpam-308 2 4 . . NOUN ejpam-308 2 5 1 1 NUM ejpam-308 2 6 , , PUNCT ejpam-308 2 7 2010 2010 NUM ejpam-308 2 8 , , PUNCT ejpam-308 2 9 13 13 NUM ejpam-308 2 10 - - SYM ejpam-308 2 11 15 15 NUM ejpam-308 2 12 issn issn PROPN ejpam-308 2 13 1307 1307 NUM ejpam-308 2 14 - - SYM ejpam-308 2 15 5543 5543 NUM ejpam-308 2 16 – – PUNCT ejpam-308 2 17 www.ejpam.com www.ejpam.com X ejpam-308 2 18 wreath wreath NOUN ejpam-308 2 19 products product NOUN ejpam-308 2 20 , , PUNCT ejpam-308 2 21 sylow sylow NOUN ejpam-308 2 22 ’s ’s PART ejpam-308 2 23 theorem theorem NOUN ejpam-308 2 24 and and CCONJ ejpam-308 2 25 fermat fermat PROPN ejpam-308 2 26 ’s ’s PART ejpam-308 2 27 little little ADJ ejpam-308 2 28 theorem theorem ADJ ejpam-308 2 29 b.sury b.sury ADJ ejpam-308 2 30 stat stat NOUN ejpam-308 2 31 - - PUNCT ejpam-308 2 32 math math NOUN ejpam-308 2 33 unit unit NOUN ejpam-308 2 34 , , PUNCT ejpam-308 2 35 indian indian PROPN ejpam-308 2 36 statistical statistical ADJ ejpam-308 2 37 institute institute NOUN ejpam-308 2 38 , , PUNCT ejpam-308 2 39 8th 8th ADJ ejpam-308 2 40 mile mile NOUN ejpam-308 2 41 mysore mysore NOUN ejpam-308 2 42 road road NOUN ejpam-308 2 43 , , PUNCT ejpam-308 2 44 bangalore bangalore NOUN ejpam-308 2 45 560 560 NUM ejpam-308 2 46 059 059 NUM ejpam-308 2 47 , , PUNCT ejpam-308 2 48 india india PROPN ejpam-308 2 49 . . PUNCT ejpam-308 3 1 abstract abstract PROPN ejpam-308 3 2 . . PUNCT ejpam-308 4 1 the the DET ejpam-308 4 2 assertion assertion NOUN ejpam-308 4 3 that that SCONJ ejpam-308 4 4 the the DET ejpam-308 4 5 number number NOUN ejpam-308 4 6 of of ADP ejpam-308 4 7 p p NOUN ejpam-308 4 8 - - PUNCT ejpam-308 4 9 sylow sylow NOUN ejpam-308 4 10 subgroups subgroup NOUN ejpam-308 4 11 in in ADP ejpam-308 4 12 a a DET ejpam-308 4 13 finite finite ADJ ejpam-308 4 14 group group NOUN ejpam-308 4 15 is be AUX ejpam-308 4 16 ≡ ≡ PROPN ejpam-308 4 17 1 1 NUM ejpam-308 4 18 mod mod PROPN ejpam-308 4 19 p p NOUN ejpam-308 4 20 , , PUNCT ejpam-308 4 21 begs beg VERB ejpam-308 4 22 the the DET ejpam-308 4 23 natural natural ADJ ejpam-308 4 24 question question NOUN ejpam-308 4 25 whether whether SCONJ ejpam-308 4 26 one one PRON ejpam-308 4 27 may may AUX ejpam-308 4 28 obtain obtain VERB ejpam-308 4 29 the the DET ejpam-308 4 30 power power NOUN ejpam-308 4 31 ap−1 ap−1 PROPN ejpam-308 4 32 ( ( PUNCT ejpam-308 4 33 for for ADP ejpam-308 4 34 any any DET ejpam-308 4 35 ( ( PUNCT ejpam-308 4 36 a a NOUN ejpam-308 4 37 , , PUNCT ejpam-308 4 38 p p NOUN ejpam-308 4 39 ) ) PUNCT ejpam-308 4 40 = = SYM ejpam-308 5 1 1 1 X ejpam-308 5 2 ) ) PUNCT ejpam-308 5 3 as as ADP ejpam-308 5 4 the the DET ejpam-308 5 5 number number NOUN ejpam-308 5 6 of of ADP ejpam-308 5 7 p p NOUN ejpam-308 5 8 - - PUNCT ejpam-308 5 9 sylow sylow NOUN ejpam-308 5 10 subgroups subgroup NOUN ejpam-308 5 11 in in ADP ejpam-308 5 12 some some DET ejpam-308 5 13 group group NOUN ejpam-308 5 14 naturally naturally ADV ejpam-308 5 15 . . PUNCT ejpam-308 6 1 indeed indeed ADV ejpam-308 6 2 , , PUNCT ejpam-308 6 3 it it PRON ejpam-308 6 4 turns turn VERB ejpam-308 6 5 out out ADP ejpam-308 6 6 to to PART ejpam-308 6 7 be be AUX ejpam-308 6 8 so so ADV ejpam-308 6 9 as as SCONJ ejpam-308 6 10 we we PRON ejpam-308 6 11 show show VERB ejpam-308 6 12 below below ADV ejpam-308 6 13 . . PUNCT ejpam-308 7 1 the the DET ejpam-308 7 2 construction construction NOUN ejpam-308 7 3 involves involve VERB ejpam-308 7 4 wreath wreath NOUN ejpam-308 7 5 products product NOUN ejpam-308 7 6 of of ADP ejpam-308 7 7 groups group NOUN ejpam-308 7 8 . . PUNCT ejpam-308 8 1 using use VERB ejpam-308 8 2 wreath wreath NOUN ejpam-308 8 3 products product NOUN ejpam-308 8 4 , , PUNCT ejpam-308 8 5 a a DET ejpam-308 8 6 different different ADJ ejpam-308 8 7 generalization generalization NOUN ejpam-308 8 8 of of ADP ejpam-308 8 9 euler euler PROPN ejpam-308 8 10 ’s ’s PART ejpam-308 8 11 congruence congruence NOUN ejpam-308 8 12 ( ( PUNCT ejpam-308 8 13 and and CCONJ ejpam-308 8 14 , , PUNCT ejpam-308 8 15 a a DET ejpam-308 8 16 fortiori fortiori X ejpam-308 8 17 , , PUNCT ejpam-308 8 18 of of ADP ejpam-308 8 19 fermat fermat PROPN ejpam-308 8 20 ’s ’s PART ejpam-308 8 21 little little ADJ ejpam-308 8 22 theorem theorem VERB ejpam-308 8 23 ) ) PUNCT ejpam-308 8 24 was be AUX ejpam-308 8 25 obtained obtain VERB ejpam-308 8 26 in in ADP ejpam-308 8 27 [ [ X ejpam-308 8 28 1 1 NUM ejpam-308 8 29 ] ] PUNCT ejpam-308 8 30 . . PUNCT ejpam-308 9 1 2000 2000 NUM ejpam-308 9 2 mathematics mathematic NOUN ejpam-308 9 3 subject subject NOUN ejpam-308 9 4 classifications classification NOUN ejpam-308 9 5 : : PUNCT ejpam-308 9 6 20d20 20d20 NUM ejpam-308 9 7 , , PUNCT ejpam-308 9 8 20d60 20d60 NUM ejpam-308 9 9 key key ADJ ejpam-308 9 10 words word NOUN ejpam-308 9 11 and and CCONJ ejpam-308 9 12 phrases phrase NOUN ejpam-308 9 13 : : PUNCT ejpam-308 9 14 wreath wreath NOUN ejpam-308 9 15 product product NOUN ejpam-308 9 16 , , PUNCT ejpam-308 9 17 sylow sylow PROPN ejpam-308 9 18 ’s ’s PART ejpam-308 9 19 theorems theorem NOUN ejpam-308 9 20 1 1 X ejpam-308 9 21 . . X ejpam-308 9 22 result result NOUN ejpam-308 9 23 given give VERB ejpam-308 9 24 two two NUM ejpam-308 9 25 groups group NOUN ejpam-308 9 26 a a PRON ejpam-308 9 27 and and CCONJ ejpam-308 9 28 b b NOUN ejpam-308 9 29 , , PUNCT ejpam-308 9 30 recall recall VERB ejpam-308 9 31 the the DET ejpam-308 9 32 restricted restricted ADJ ejpam-308 9 33 wreath wreath NOUN ejpam-308 9 34 product product NOUN ejpam-308 9 35 of of ADP ejpam-308 9 36 a a PRON ejpam-308 9 37 by by ADP ejpam-308 9 38 b b PROPN ejpam-308 9 39 ( ( PUNCT ejpam-308 9 40 written write VERB ejpam-308 9 41 a a DET ejpam-308 9 42 ≀ ≀ NOUN ejpam-308 9 43 b b NOUN ejpam-308 9 44 ) ) PUNCT ejpam-308 9 45 . . PUNCT ejpam-308 10 1 this this PRON ejpam-308 10 2 is be AUX ejpam-308 10 3 the the DET ejpam-308 10 4 semidirect semidirect NOUN ejpam-308 10 5 product product NOUN ejpam-308 10 6 group group NOUN ejpam-308 10 7 b b PROPN ejpam-308 10 8 ∝ ∝ PROPN ejpam-308 10 9 ã ã PROPN ejpam-308 10 10 where where SCONJ ejpam-308 10 11 ã ã PROPN ejpam-308 10 12 = = SYM ejpam-308 10 13 ⊕b∈bab ⊕b∈bab PROPN ejpam-308 10 14 with with ADP ejpam-308 10 15 each each DET ejpam-308 10 16 ab ab NOUN ejpam-308 10 17 = = PUNCT ejpam-308 10 18 a a PROPN ejpam-308 10 19 and and CCONJ ejpam-308 10 20 b b NOUN ejpam-308 10 21 acts act NOUN ejpam-308 10 22 on on ADP ejpam-308 10 23 the the DET ejpam-308 10 24 indexing indexing NOUN ejpam-308 10 25 set set VERB ejpam-308 10 26 b b PROPN ejpam-308 10 27 of of ADP ejpam-308 10 28 ã ã PROPN ejpam-308 10 29 by by ADP ejpam-308 10 30 right right ADJ ejpam-308 10 31 multiplication multiplication NOUN ejpam-308 10 32 . . PUNCT ejpam-308 11 1 we we PRON ejpam-308 11 2 write write VERB ejpam-308 11 3 any any DET ejpam-308 11 4 element element NOUN ejpam-308 11 5 of of ADP ejpam-308 11 6 a a DET ejpam-308 11 7 ≀ ≀ PROPN ejpam-308 11 8 b b NOUN ejpam-308 11 9 in in ADP ejpam-308 11 10 a a DET ejpam-308 11 11 canonical canonical ADJ ejpam-308 11 12 form form NOUN ejpam-308 11 13 as as ADP ejpam-308 11 14 σa1 σa1 PROPN ejpam-308 11 15 ( ( PUNCT ejpam-308 11 16 b1 b1 NOUN ejpam-308 11 17 ) ) PUNCT ejpam-308 11 18 · · PUNCT ejpam-308 11 19 · · PUNCT ejpam-308 12 1 · · PUNCT ejpam-308 12 2 σar σar X ejpam-308 12 3 ( ( PUNCT ejpam-308 12 4 br)τ(b br)τ(b NOUN ejpam-308 12 5 ) ) PUNCT ejpam-308 12 6 where where SCONJ ejpam-308 12 7 ai ai VERB ejpam-308 12 8 ∈ ∈ PROPN ejpam-308 12 9 a a PRON ejpam-308 12 10 ; ; PUNCT ejpam-308 12 11 bi bi NOUN ejpam-308 12 12 , , PUNCT ejpam-308 12 13 b b PROPN ejpam-308 12 14 ∈ ∈ PROPN ejpam-308 12 15 b. b. PROPN ejpam-308 12 16 thus thus ADV ejpam-308 12 17 , , PUNCT ejpam-308 12 18 two two NUM ejpam-308 12 19 elements element NOUN ejpam-308 12 20 σa1 σa1 VERB ejpam-308 12 21 ( ( PUNCT ejpam-308 12 22 b1 b1 NOUN ejpam-308 12 23 ) ) PUNCT ejpam-308 12 24 and and CCONJ ejpam-308 12 25 σa2 σa2 NOUN ejpam-308 12 26 ( ( PUNCT ejpam-308 12 27 b2 b2 NOUN ejpam-308 12 28 ) ) PUNCT ejpam-308 12 29 commute commute NOUN ejpam-308 12 30 if if SCONJ ejpam-308 12 31 b1 b1 PROPN ejpam-308 12 32 6= 6= SYM ejpam-308 12 33 b2 b2 NOUN ejpam-308 12 34 . . PUNCT ejpam-308 13 1 also also ADV ejpam-308 13 2 , , PUNCT ejpam-308 13 3 the the DET ejpam-308 13 4 product product NOUN ejpam-308 13 5 σa1 σa1 NOUN ejpam-308 13 6 ( ( PUNCT ejpam-308 13 7 b)σa2 b)σa2 X ejpam-308 13 8 ( ( PUNCT ejpam-308 13 9 b b NOUN ejpam-308 13 10 ) ) PUNCT ejpam-308 13 11 = = PUNCT ejpam-308 14 1 σa1a2 σa1a2 ADJ ejpam-308 14 2 ( ( PUNCT ejpam-308 14 3 b b NOUN ejpam-308 14 4 ) ) PUNCT ejpam-308 14 5 . . PUNCT ejpam-308 15 1 finally finally ADV ejpam-308 15 2 , , PUNCT ejpam-308 15 3 τ(b)σa(c)τ(b τ(b)σa(c)τ(b NOUN ejpam-308 15 4 ) ) PUNCT ejpam-308 15 5 −1 −1 NOUN ejpam-308 15 6 = = SYM ejpam-308 15 7 σa(cb σa(cb PROPN ejpam-308 15 8 ) ) PUNCT ejpam-308 15 9 . . PUNCT ejpam-308 16 1 we we PRON ejpam-308 16 2 prove prove VERB ejpam-308 16 3 : : PUNCT ejpam-308 16 4 theorem theorem NOUN ejpam-308 16 5 1 1 NUM ejpam-308 16 6 . . PUNCT ejpam-308 17 1 let let VERB ejpam-308 17 2 |b| |b| PROPN ejpam-308 17 3 = = SYM ejpam-308 17 4 p p X ejpam-308 17 5 , , PUNCT ejpam-308 17 6 a a DET ejpam-308 17 7 prime prime NOUN ejpam-308 17 8 and and CCONJ ejpam-308 17 9 , , PUNCT ejpam-308 17 10 ( ( PUNCT ejpam-308 17 11 |a| |a| NOUN ejpam-308 17 12 , , PUNCT ejpam-308 17 13 p p NOUN ejpam-308 17 14 ) ) PUNCT ejpam-308 17 15 = = SYM ejpam-308 18 1 1 1 X ejpam-308 18 2 . . PUNCT ejpam-308 19 1 then then ADV ejpam-308 19 2 , , PUNCT ejpam-308 19 3 the the DET ejpam-308 19 4 number number NOUN ejpam-308 19 5 of of ADP ejpam-308 19 6 p p NOUN ejpam-308 19 7 - - PUNCT ejpam-308 19 8 sylow sylow NOUN ejpam-308 19 9 subgroups subgroup NOUN ejpam-308 19 10 in in ADP ejpam-308 19 11 the the DET ejpam-308 19 12 wreath wreath NOUN ejpam-308 19 13 product product NOUN ejpam-308 19 14 a a DET ejpam-308 19 15 ≀ ≀ PROPN ejpam-308 19 16 b b PROPN ejpam-308 19 17 is be AUX ejpam-308 19 18 |a|p−1 |a|p−1 ADJ ejpam-308 19 19 . . PUNCT ejpam-308 20 1 thus thus ADV ejpam-308 20 2 , , PUNCT ejpam-308 20 3 |a|p−1 |a|p−1 ADJ ejpam-308 20 4 ≡ ≡ PROPN ejpam-308 20 5 1 1 NUM ejpam-308 20 6 mod mod PROPN ejpam-308 20 7 p. p. NOUN ejpam-308 20 8 to to PART ejpam-308 20 9 prove prove VERB ejpam-308 20 10 the the DET ejpam-308 20 11 theorem theorem NOUN ejpam-308 20 12 , , PUNCT ejpam-308 20 13 we we PRON ejpam-308 20 14 shall shall AUX ejpam-308 20 15 use use VERB ejpam-308 20 16 a a DET ejpam-308 20 17 lemma lemma PROPN ejpam-308 20 18 on on ADP ejpam-308 20 19 the the DET ejpam-308 20 20 wreath wreath NOUN ejpam-308 20 21 product product NOUN ejpam-308 20 22 a a DET ejpam-308 20 23 ≀ ≀ PROPN ejpam-308 20 24 b b PROPN ejpam-308 20 25 of of ADP ejpam-308 20 26 two two NUM ejpam-308 20 27 arbitrary arbitrary ADJ ejpam-308 20 28 finite finite ADJ ejpam-308 20 29 groups group NOUN ejpam-308 20 30 . . PUNCT ejpam-308 21 1 let let VERB ejpam-308 21 2 us we PRON ejpam-308 21 3 denote denote VERB ejpam-308 21 4 by by ADP ejpam-308 21 5 c c PROPN ejpam-308 21 6 the the DET ejpam-308 21 7 subgroup subgroup NOUN ejpam-308 21 8 c c PROPN ejpam-308 21 9 = = PRON ejpam-308 21 10 { { PUNCT ejpam-308 21 11 σa(b1 σa(b1 NOUN ejpam-308 21 12 ) ) PUNCT ejpam-308 21 13 · · PUNCT ejpam-308 21 14 · · PUNCT ejpam-308 21 15 · · PUNCT ejpam-308 21 16 σa(bn σa(bn PROPN ejpam-308 21 17 ) ) PUNCT ejpam-308 21 18 : : PUNCT ejpam-308 21 19 a a DET ejpam-308 21 20 ∈ ∈ PROPN ejpam-308 21 21 a a X ejpam-308 21 22 } } PUNCT ejpam-308 21 23 where where SCONJ ejpam-308 21 24 b b X ejpam-308 21 25 = = SYM ejpam-308 21 26 { { PUNCT ejpam-308 21 27 b1 b1 PROPN ejpam-308 21 28 , , PUNCT ejpam-308 21 29 · · PUNCT ejpam-308 21 30 · · PUNCT ejpam-308 21 31 · · PUNCT ejpam-308 21 32 , , PUNCT ejpam-308 21 33 bn bn ADJ ejpam-308 21 34 } } PUNCT ejpam-308 21 35 . . PUNCT ejpam-308 22 1 note note VERB ejpam-308 22 2 that that SCONJ ejpam-308 22 3 all all DET ejpam-308 22 4 the the DET ejpam-308 22 5 elements element NOUN ejpam-308 22 6 τ(b τ(b NOUN ejpam-308 22 7 ) ) PUNCT ejpam-308 22 8 for for ADP ejpam-308 22 9 b b PROPN ejpam-308 22 10 ∈ ∈ PROPN ejpam-308 22 11 b b PROPN ejpam-308 22 12 commute commute NOUN ejpam-308 22 13 element element NOUN ejpam-308 22 14 - - ADJ ejpam-308 22 15 wise wise ADJ ejpam-308 22 16 with with ADP ejpam-308 22 17 this this DET ejpam-308 22 18 subgroup subgroup NOUN ejpam-308 22 19 . . PUNCT ejpam-308 23 1 email email NOUN ejpam-308 23 2 address address NOUN ejpam-308 23 3 : : PUNCT ejpam-308 23 4 sury sury PROPN ejpam-308 23 5 � � PROPN ejpam-308 23 6 isibang.a isibang.a PROPN ejpam-308 23 7 .in .in PUNCT ejpam-308 23 8 ( ( PUNCT ejpam-308 23 9 b. b. PROPN ejpam-308 23 10 sury sury PROPN ejpam-308 23 11 ) ) PUNCT ejpam-308 23 12 http://www.ejpam.com http://www.ejpam.com X ejpam-308 24 1 13 13 NUM ejpam-308 25 1 c c X ejpam-308 25 2 © © PROPN ejpam-308 25 3 2009 2009 NUM ejpam-308 25 4 ejpam ejpam NOUN ejpam-308 25 5 all all DET ejpam-308 25 6 rights right NOUN ejpam-308 25 7 reserved reserve VERB ejpam-308 25 8 . . PUNCT ejpam-308 26 1 b. b. PROPN ejpam-308 26 2 sury sury PROPN ejpam-308 26 3 / / SYM ejpam-308 26 4 eur eur PROPN ejpam-308 26 5 . . PUNCT ejpam-308 27 1 j. j. PROPN ejpam-308 27 2 pure pure PROPN ejpam-308 27 3 appl appl PROPN ejpam-308 27 4 . . PROPN ejpam-308 27 5 math math PROPN ejpam-308 27 6 , , PUNCT ejpam-308 27 7 3 3 NUM ejpam-308 27 8 ( ( PUNCT ejpam-308 27 9 2010 2010 NUM ejpam-308 27 10 ) ) PUNCT ejpam-308 27 11 , , PUNCT ejpam-308 27 12 13 13 NUM ejpam-308 27 13 - - SYM ejpam-308 27 14 15 15 NUM ejpam-308 27 15 14 14 NUM ejpam-308 27 16 lemma lemma PROPN ejpam-308 27 17 1 1 NUM ejpam-308 27 18 . . PUNCT ejpam-308 28 1 let let VERB ejpam-308 28 2 a a DET ejpam-308 28 3 , , PUNCT ejpam-308 28 4 b b PROPN ejpam-308 28 5 be be AUX ejpam-308 28 6 finite finite ADJ ejpam-308 28 7 groups group NOUN ejpam-308 28 8 . . PUNCT ejpam-308 29 1 then then ADV ejpam-308 29 2 , , PUNCT ejpam-308 29 3 the the DET ejpam-308 29 4 normalizer normalizer NOUN ejpam-308 29 5 of of ADP ejpam-308 29 6 the the DET ejpam-308 29 7 subgroup subgroup PROPN ejpam-308 29 8 b b PROPN ejpam-308 29 9 in in ADP ejpam-308 29 10 a≀b a≀b PROPN ejpam-308 29 11 equals equal VERB ejpam-308 29 12 c⊕b c⊕b NOUN ejpam-308 29 13 . . PUNCT ejpam-308 30 1 proof proof NOUN ejpam-308 30 2 . . PUNCT ejpam-308 31 1 indeed indeed ADV ejpam-308 31 2 , , PUNCT ejpam-308 31 3 if if SCONJ ejpam-308 31 4 σa1 σa1 X ejpam-308 31 5 ( ( PUNCT ejpam-308 31 6 b1 b1 NOUN ejpam-308 31 7 ) ) PUNCT ejpam-308 31 8 · · PUNCT ejpam-308 31 9 · · PUNCT ejpam-308 32 1 · · PUNCT ejpam-308 32 2 σar σar X ejpam-308 32 3 ( ( PUNCT ejpam-308 32 4 br)τ(b0 br)τ(b0 NOUN ejpam-308 32 5 ) ) PUNCT ejpam-308 32 6 is be AUX ejpam-308 32 7 in in ADP ejpam-308 32 8 the the DET ejpam-308 32 9 normalizer normalizer NOUN ejpam-308 32 10 of of ADP ejpam-308 32 11 b b PROPN ejpam-308 32 12 , , PUNCT ejpam-308 32 13 we we PRON ejpam-308 32 14 have have VERB ejpam-308 32 15 for for ADP ejpam-308 32 16 each each DET ejpam-308 32 17 b b PROPN ejpam-308 32 18 ∈ ∈ PROPN ejpam-308 32 19 b b PROPN ejpam-308 32 20 , , PUNCT ejpam-308 32 21 some some DET ejpam-308 32 22 b′ b′ NUM ejpam-308 32 23 ∈ ∈ PROPN ejpam-308 32 24 b b NOUN ejpam-308 32 25 so so ADV ejpam-308 32 26 that that DET ejpam-308 32 27 σa1 σa1 PROPN ejpam-308 32 28 ( ( PUNCT ejpam-308 32 29 b1 b1 PROPN ejpam-308 32 30 ) ) PUNCT ejpam-308 32 31 · · PUNCT ejpam-308 32 32 · · PUNCT ejpam-308 33 1 · · PUNCT ejpam-308 33 2 σar σar X ejpam-308 33 3 ( ( PUNCT ejpam-308 33 4 br)τ(b0)τ(b br)τ(b0)τ(b NOUN ejpam-308 33 5 ) ) PUNCT ejpam-308 33 6 = = SYM ejpam-308 33 7 τ(b τ(b PROPN ejpam-308 33 8 ′)σa1 ′)σa1 PROPN ejpam-308 33 9 ( ( PUNCT ejpam-308 33 10 b1 b1 PROPN ejpam-308 33 11 ) ) PUNCT ejpam-308 33 12 · · PUNCT ejpam-308 33 13 · · PUNCT ejpam-308 33 14 · · PUNCT ejpam-308 33 15 σar σar X ejpam-308 33 16 ( ( PUNCT ejpam-308 33 17 br)τ(b0 br)τ(b0 NOUN ejpam-308 33 18 ) ) PUNCT ejpam-308 33 19 = = SYM ejpam-308 34 1 σa1 σa1 PROPN ejpam-308 34 2 ( ( PUNCT ejpam-308 34 3 b1 b1 NOUN ejpam-308 34 4 b′ b′ NUM ejpam-308 34 5 ) ) PUNCT ejpam-308 34 6 · · PUNCT ejpam-308 34 7 · · PUNCT ejpam-308 34 8 · · PUNCT ejpam-308 34 9 σar σar X ejpam-308 34 10 ( ( PUNCT ejpam-308 34 11 br br PROPN ejpam-308 34 12 b′)τ(b′b0 b′)τ(b′b0 PROPN ejpam-308 34 13 ) ) PUNCT ejpam-308 34 14 . . PUNCT ejpam-308 35 1 so so ADV ejpam-308 35 2 b0 b0 VERB ejpam-308 35 3 bb−1 bb−1 NOUN ejpam-308 35 4 0 0 NUM ejpam-308 36 1 = = SYM ejpam-308 36 2 b′ b′ NUM ejpam-308 36 3 and and CCONJ ejpam-308 36 4 σa1 σa1 PROPN ejpam-308 36 5 ( ( PUNCT ejpam-308 36 6 b1 b1 NOUN ejpam-308 36 7 ) ) PUNCT ejpam-308 36 8 · · PUNCT ejpam-308 36 9 · · PUNCT ejpam-308 36 10 · · PUNCT ejpam-308 36 11 σar σar X ejpam-308 36 12 ( ( PUNCT ejpam-308 36 13 br br NOUN ejpam-308 36 14 ) ) PUNCT ejpam-308 37 1 = = SYM ejpam-308 37 2 σa1 σa1 PROPN ejpam-308 37 3 ( ( PUNCT ejpam-308 37 4 b1 b1 NOUN ejpam-308 37 5 b′ b′ NUM ejpam-308 37 6 ) ) PUNCT ejpam-308 37 7 · · PUNCT ejpam-308 37 8 · · PUNCT ejpam-308 37 9 · · PUNCT ejpam-308 37 10 σar σar X ejpam-308 37 11 ( ( PUNCT ejpam-308 37 12 br br NOUN ejpam-308 37 13 b′ b′ NUM ejpam-308 37 14 ) ) PUNCT ejpam-308 38 1 = = SYM ejpam-308 38 2 σa1 σa1 PROPN ejpam-308 38 3 ( ( PUNCT ejpam-308 38 4 b1 b1 NOUN ejpam-308 38 5 b0 b0 ADP ejpam-308 38 6 bb−1 bb−1 NOUN ejpam-308 38 7 0 0 NUM ejpam-308 38 8 ) ) PUNCT ejpam-308 38 9 · · PUNCT ejpam-308 38 10 · · PUNCT ejpam-308 39 1 · · PUNCT ejpam-308 39 2 σar σar AUX ejpam-308 39 3 ( ( PUNCT ejpam-308 39 4 br br NOUN ejpam-308 39 5 b0 b0 VERB ejpam-308 39 6 bb−1 bb−1 NOUN ejpam-308 39 7 0 0 NUM ejpam-308 39 8 ) ) PUNCT ejpam-308 39 9 . . PUNCT ejpam-308 40 1 as as SCONJ ejpam-308 40 2 b b PROPN ejpam-308 40 3 can can AUX ejpam-308 40 4 take take VERB ejpam-308 40 5 any any DET ejpam-308 40 6 value value NOUN ejpam-308 40 7 in in ADP ejpam-308 40 8 b b NOUN ejpam-308 40 9 , , PUNCT ejpam-308 40 10 and and CCONJ ejpam-308 40 11 bi(b0 bi(b0 ADP ejpam-308 40 12 b−1 b−1 NOUN ejpam-308 40 13 0 0 NUM ejpam-308 40 14 ) ) PUNCT ejpam-308 40 15 ; ; PUNCT ejpam-308 40 16 i i PRON ejpam-308 40 17 ≤ ≤ NOUN ejpam-308 40 18 r r NOUN ejpam-308 40 19 are be AUX ejpam-308 40 20 distinct distinct ADJ ejpam-308 40 21 elements element NOUN ejpam-308 40 22 , , PUNCT ejpam-308 40 23 we we PRON ejpam-308 40 24 must must AUX ejpam-308 40 25 have have VERB ejpam-308 40 26 b b NOUN ejpam-308 40 27 = = SYM ejpam-308 40 28 { { PUNCT ejpam-308 40 29 b1 b1 PROPN ejpam-308 40 30 , , PUNCT ejpam-308 40 31 · · PUNCT ejpam-308 40 32 · · PUNCT ejpam-308 40 33 · · PUNCT ejpam-308 40 34 , , PUNCT ejpam-308 40 35 br br PROPN ejpam-308 40 36 } } PUNCT ejpam-308 40 37 . . PUNCT ejpam-308 41 1 moreover moreover ADV ejpam-308 41 2 , , PUNCT ejpam-308 41 3 looking look VERB ejpam-308 41 4 at at ADP ejpam-308 41 5 a a DET ejpam-308 41 6 b b NOUN ejpam-308 41 7 such such ADJ ejpam-308 41 8 that that PRON ejpam-308 41 9 bi(b0 bi(b0 ADJ ejpam-308 41 10 b−1 b−1 NOUN ejpam-308 41 11 0 0 NUM ejpam-308 41 12 ) ) PUNCT ejpam-308 42 1 = = SYM ejpam-308 42 2 b b X ejpam-308 43 1 j j PROPN ejpam-308 43 2 , , PUNCT ejpam-308 43 3 we we PRON ejpam-308 43 4 must must AUX ejpam-308 43 5 have have AUX ejpam-308 43 6 ai ai VERB ejpam-308 43 7 = = PROPN ejpam-308 43 8 a a DET ejpam-308 43 9 j. j. PROPN ejpam-308 43 10 thus thus ADV ejpam-308 43 11 , , PUNCT ejpam-308 43 12 σa1 σa1 INTJ ejpam-308 43 13 ( ( PUNCT ejpam-308 43 14 b1 b1 PROPN ejpam-308 43 15 ) ) PUNCT ejpam-308 43 16 · · PUNCT ejpam-308 43 17 · · PUNCT ejpam-308 44 1 · · PUNCT ejpam-308 44 2 σar σar X ejpam-308 44 3 ( ( PUNCT ejpam-308 44 4 br br NOUN ejpam-308 44 5 ) ) PUNCT ejpam-308 44 6 ∈ ∈ PROPN ejpam-308 44 7 c c NOUN ejpam-308 44 8 . . PUNCT ejpam-308 45 1 this this PRON ejpam-308 45 2 proves prove VERB ejpam-308 45 3 the the DET ejpam-308 45 4 lemma lemma PROPN ejpam-308 45 5 . . PUNCT ejpam-308 46 1 proof proof NOUN ejpam-308 46 2 . . PUNCT ejpam-308 47 1 [ [ X ejpam-308 47 2 theorem theorem NOUN ejpam-308 47 3 1 1 NUM ejpam-308 47 4 ] ] PUNCT ejpam-308 47 5 here here ADV ejpam-308 47 6 , , PUNCT ejpam-308 47 7 since since SCONJ ejpam-308 47 8 p p NOUN ejpam-308 47 9 is be AUX ejpam-308 47 10 the the DET ejpam-308 47 11 highest high ADJ ejpam-308 47 12 power power NOUN ejpam-308 47 13 of of ADP ejpam-308 47 14 p p NOUN ejpam-308 47 15 dividing divide VERB ejpam-308 47 16 the the DET ejpam-308 47 17 order order NOUN ejpam-308 47 18 of of ADP ejpam-308 47 19 a a DET ejpam-308 47 20 ≀ ≀ PROPN ejpam-308 47 21 b b NOUN ejpam-308 47 22 , , PUNCT ejpam-308 47 23 the the DET ejpam-308 47 24 subgroup subgroup PROPN ejpam-308 47 25 b b PROPN ejpam-308 47 26 is be AUX ejpam-308 47 27 a a DET ejpam-308 47 28 p p ADJ ejpam-308 47 29 - - PUNCT ejpam-308 47 30 sylow sylow NOUN ejpam-308 47 31 subgroup subgroup NOUN ejpam-308 47 32 . . PUNCT ejpam-308 48 1 by by ADP ejpam-308 48 2 lemma lemma PROPN ejpam-308 48 3 1 1 NUM ejpam-308 48 4 , , PUNCT ejpam-308 48 5 the the DET ejpam-308 48 6 normalizer normalizer PROPN ejpam-308 48 7 n(b n(b PROPN ejpam-308 48 8 ) ) PUNCT ejpam-308 48 9 of of ADP ejpam-308 48 10 b b PROPN ejpam-308 48 11 has have VERB ejpam-308 48 12 order order NOUN ejpam-308 48 13 equal equal ADJ ejpam-308 48 14 to to ADP ejpam-308 48 15 p|a| p|a| ADJ ejpam-308 48 16 . . PUNCT ejpam-308 49 1 since since SCONJ ejpam-308 49 2 |a |a NOUN ejpam-308 49 3 ≀ ≀ CCONJ ejpam-308 49 4 b| b| PROPN ejpam-308 49 5 = = SYM ejpam-308 49 6 p|a|p p|a|p PROPN ejpam-308 49 7 , , PUNCT ejpam-308 49 8 we we PRON ejpam-308 49 9 have have VERB ejpam-308 49 10 [ [ PUNCT ejpam-308 49 11 a a X ejpam-308 49 12 ≀ ≀ X ejpam-308 49 13 b b NOUN ejpam-308 49 14 : : PUNCT ejpam-308 49 15 n(b n(b PROPN ejpam-308 49 16 ) ) PUNCT ejpam-308 49 17 ] ] PUNCT ejpam-308 50 1 = = PUNCT ejpam-308 50 2 |a|p−1 |a|p−1 X ejpam-308 50 3 . . PUNCT ejpam-308 51 1 by by ADP ejpam-308 51 2 the the DET ejpam-308 51 3 second second ADJ ejpam-308 51 4 sylow sylow NOUN ejpam-308 51 5 theorem theorem VERB ejpam-308 51 6 , , PUNCT ejpam-308 51 7 the the DET ejpam-308 51 8 number number NOUN ejpam-308 51 9 of of ADP ejpam-308 51 10 p p NOUN ejpam-308 51 11 - - PUNCT ejpam-308 51 12 sylow sylow NOUN ejpam-308 51 13 subgroups subgroup NOUN ejpam-308 51 14 equals equal VERB ejpam-308 51 15 the the DET ejpam-308 51 16 index index NOUN ejpam-308 51 17 of of ADP ejpam-308 51 18 the the DET ejpam-308 51 19 normalizer normalizer NOUN ejpam-308 51 20 . . PUNCT ejpam-308 52 1 by by ADP ejpam-308 52 2 the the DET ejpam-308 52 3 third third ADJ ejpam-308 52 4 sylow sylow NOUN ejpam-308 52 5 theorem theorem VERB ejpam-308 52 6 , , PUNCT ejpam-308 52 7 this this DET ejpam-308 52 8 number number NOUN ejpam-308 52 9 is be AUX ejpam-308 52 10 congruent congruent ADJ ejpam-308 52 11 to to ADP ejpam-308 52 12 1 1 NUM ejpam-308 52 13 mod mod NOUN ejpam-308 52 14 p. p. PROPN ejpam-308 52 15 lemma lemma PROPN ejpam-308 53 1 2 2 X ejpam-308 53 2 . . PUNCT ejpam-308 53 3 let let VERB ejpam-308 53 4 a a DET ejpam-308 53 5 , , PUNCT ejpam-308 53 6 b b PROPN ejpam-308 53 7 be be AUX ejpam-308 53 8 finite finite ADJ ejpam-308 53 9 solvable solvable ADJ ejpam-308 53 10 groups group NOUN ejpam-308 53 11 of of ADP ejpam-308 53 12 orders order NOUN ejpam-308 53 13 a a PRON ejpam-308 53 14 , , PUNCT ejpam-308 53 15 b b NOUN ejpam-308 53 16 with with ADP ejpam-308 53 17 ( ( PUNCT ejpam-308 53 18 a a DET ejpam-308 53 19 , , PUNCT ejpam-308 53 20 b b NOUN ejpam-308 53 21 ) ) PUNCT ejpam-308 53 22 = = SYM ejpam-308 54 1 1 1 X ejpam-308 54 2 . . PUNCT ejpam-308 55 1 then then ADV ejpam-308 55 2 , , PUNCT ejpam-308 55 3 the the DET ejpam-308 55 4 subgroups subgroup NOUN ejpam-308 55 5 of of ADP ejpam-308 55 6 order order NOUN ejpam-308 55 7 b b PROPN ejpam-308 55 8 in in ADP ejpam-308 55 9 a a DET ejpam-308 55 10 ≀ ≀ NOUN ejpam-308 55 11 b b NOUN ejpam-308 55 12 are be AUX ejpam-308 55 13 conjugate conjugate ADJ ejpam-308 55 14 and and CCONJ ejpam-308 55 15 , , PUNCT ejpam-308 55 16 are be AUX ejpam-308 55 17 ab−1 ab−1 NOUN ejpam-308 55 18 in in ADP ejpam-308 55 19 number number NOUN ejpam-308 55 20 . . PUNCT ejpam-308 56 1 proof proof NOUN ejpam-308 56 2 . . PUNCT ejpam-308 57 1 if if SCONJ ejpam-308 57 2 g g PROPN ejpam-308 57 3 is be AUX ejpam-308 57 4 a a DET ejpam-308 57 5 solvable solvable ADJ ejpam-308 57 6 group group NOUN ejpam-308 57 7 of of ADP ejpam-308 57 8 order order NOUN ejpam-308 57 9 mn mn PROPN ejpam-308 57 10 , , PUNCT ejpam-308 57 11 with with ADP ejpam-308 57 12 ( ( PUNCT ejpam-308 57 13 m m PROPN ejpam-308 57 14 , , PUNCT ejpam-308 57 15 n n CCONJ ejpam-308 57 16 ) ) PUNCT ejpam-308 57 17 = = SYM ejpam-308 57 18 1 1 NUM ejpam-308 57 19 , , PUNCT ejpam-308 57 20 then then ADV ejpam-308 57 21 it it PRON ejpam-308 57 22 is be AUX ejpam-308 57 23 well well ADV ejpam-308 57 24 - - PUNCT ejpam-308 57 25 known know VERB ejpam-308 57 26 that that SCONJ ejpam-308 57 27 g g PROPN ejpam-308 57 28 has have VERB ejpam-308 57 29 subgroups subgroup NOUN ejpam-308 57 30 of of ADP ejpam-308 57 31 order order NOUN ejpam-308 57 32 m m VERB ejpam-308 57 33 which which PRON ejpam-308 57 34 are be AUX ejpam-308 57 35 pairwise pairwise NOUN ejpam-308 57 36 conjugate conjugate NOUN ejpam-308 57 37 . . PUNCT ejpam-308 58 1 now now ADV ejpam-308 58 2 a≀b a≀b PROPN ejpam-308 58 3 has have VERB ejpam-308 58 4 ã ã PROPN ejpam-308 58 5 as as ADP ejpam-308 58 6 a a DET ejpam-308 58 7 normal normal ADJ ejpam-308 58 8 subgroup subgroup NOUN ejpam-308 58 9 and and CCONJ ejpam-308 58 10 the the DET ejpam-308 58 11 quotient quotient NOUN ejpam-308 58 12 is be AUX ejpam-308 58 13 isomorphic isomorphic ADJ ejpam-308 58 14 to to AUX ejpam-308 58 15 b. b. PROPN ejpam-308 58 16 as as SCONJ ejpam-308 58 17 a a PRON ejpam-308 58 18 is be AUX ejpam-308 58 19 solvable solvable ADJ ejpam-308 58 20 , , PUNCT ejpam-308 58 21 so so ADV ejpam-308 58 22 is be AUX ejpam-308 58 23 the the DET ejpam-308 58 24 group group NOUN ejpam-308 58 25 ã. ã. PROPN ejpam-308 58 26 hence hence ADV ejpam-308 58 27 , , PUNCT ejpam-308 58 28 a≀b a≀b PROPN ejpam-308 58 29 is be AUX ejpam-308 58 30 solvable solvable ADJ ejpam-308 58 31 as as SCONJ ejpam-308 58 32 both both CCONJ ejpam-308 58 33 ã ã PROPN ejpam-308 58 34 and and CCONJ ejpam-308 58 35 b b NOUN ejpam-308 58 36 are be AUX ejpam-308 58 37 solvable solvable ADJ ejpam-308 58 38 . . PUNCT ejpam-308 59 1 thus thus ADV ejpam-308 59 2 , , PUNCT ejpam-308 59 3 the the DET ejpam-308 59 4 subgroups subgroup NOUN ejpam-308 59 5 of of ADP ejpam-308 59 6 order order NOUN ejpam-308 59 7 b b X ejpam-308 59 8 in in ADP ejpam-308 59 9 it it PRON ejpam-308 59 10 are be AUX ejpam-308 59 11 pairwise pairwise NOUN ejpam-308 59 12 conjugate conjugate ADJ ejpam-308 59 13 and and CCONJ ejpam-308 59 14 are be AUX ejpam-308 59 15 , , PUNCT ejpam-308 59 16 thus thus ADV ejpam-308 59 17 , , PUNCT ejpam-308 59 18 [ [ X ejpam-308 59 19 a a X ejpam-308 59 20 ≀ ≀ X ejpam-308 59 21 b b NOUN ejpam-308 59 22 : : PUNCT ejpam-308 59 23 n(b n(b PROPN ejpam-308 59 24 ) ) PUNCT ejpam-308 59 25 ] ] PUNCT ejpam-308 59 26 in in ADP ejpam-308 59 27 number number NOUN ejpam-308 59 28 . . PUNCT ejpam-308 60 1 this this DET ejpam-308 60 2 index index NOUN ejpam-308 60 3 is be AUX ejpam-308 60 4 ab ab PROPN ejpam-308 60 5 b b PROPN ejpam-308 60 6 / / SYM ejpam-308 60 7 ab ab NOUN ejpam-308 60 8 = = SYM ejpam-308 60 9 ab−1 ab−1 PROPN ejpam-308 60 10 . . PUNCT ejpam-308 61 1 2 2 NUM ejpam-308 61 2 . . X ejpam-308 61 3 remarks remark VERB ejpam-308 61 4 the the DET ejpam-308 61 5 wreath wreath NOUN ejpam-308 61 6 product product NOUN ejpam-308 61 7 of of ADP ejpam-308 61 8 finite finite ADJ ejpam-308 61 9 groups group NOUN ejpam-308 61 10 was be AUX ejpam-308 61 11 considered consider VERB ejpam-308 61 12 in in ADP ejpam-308 61 13 [ [ X ejpam-308 61 14 1 1 X ejpam-308 61 15 ] ] PUNCT ejpam-308 61 16 also also ADV ejpam-308 61 17 , , PUNCT ejpam-308 61 18 where where SCONJ ejpam-308 61 19 a a DET ejpam-308 61 20 different different ADJ ejpam-308 61 21 generalization generalization NOUN ejpam-308 61 22 of of ADP ejpam-308 61 23 euler euler PROPN ejpam-308 61 24 ’s ’s PART ejpam-308 61 25 congruence congruence NOUN ejpam-308 61 26 dropped drop VERB ejpam-308 61 27 out out ADP ejpam-308 61 28 as as ADP ejpam-308 61 29 a a DET ejpam-308 61 30 byproduct byproduct NOUN ejpam-308 61 31 . . PUNCT ejpam-308 62 1 a a DET ejpam-308 62 2 particular particular ADJ ejpam-308 62 3 case case NOUN ejpam-308 62 4 is be AUX ejpam-308 62 5 : : PUNCT ejpam-308 62 6 let let VERB ejpam-308 62 7 a a DET ejpam-308 62 8 , , PUNCT ejpam-308 62 9 b b PROPN ejpam-308 62 10 be be AUX ejpam-308 62 11 finite finite ADJ ejpam-308 62 12 abelian abelian ADJ ejpam-308 62 13 groups group NOUN ejpam-308 62 14 of of ADP ejpam-308 62 15 orders order NOUN ejpam-308 62 16 a a PRON ejpam-308 62 17 , , PUNCT ejpam-308 62 18 b b NOUN ejpam-308 62 19 respectively respectively ADV ejpam-308 62 20 . . PUNCT ejpam-308 63 1 then then ADV ejpam-308 63 2 , , PUNCT ejpam-308 63 3 the the DET ejpam-308 63 4 number number NOUN ejpam-308 63 5 of of ADP ejpam-308 63 6 conjugacy conjugacy ADJ ejpam-308 63 7 classes class NOUN ejpam-308 63 8 in in ADP ejpam-308 63 9 the the DET ejpam-308 63 10 wreath wreath NOUN ejpam-308 63 11 product product NOUN ejpam-308 63 12 a≀b a≀b PROPN ejpam-308 63 13 is be AUX ejpam-308 63 14 1 1 NUM ejpam-308 63 15 b b NUM ejpam-308 63 16 ∑ ∑ PROPN ejpam-308 63 17 s s PROPN ejpam-308 63 18 , , PUNCT ejpam-308 63 19 t∈b t∈b NOUN ejpam-308 63 20 a[b: > X ejpam-308 63 24 ] ] PUNCT ejpam-308 63 25 . . PUNCT ejpam-308 64 1 in in ADP ejpam-308 64 2 particular particular ADJ ejpam-308 64 3 , , PUNCT ejpam-308 64 4 when when SCONJ ejpam-308 64 5 b b NOUN ejpam-308 64 6 is be AUX ejpam-308 64 7 cyclic cyclic ADJ ejpam-308 64 8 , , PUNCT ejpam-308 64 9 this this DET ejpam-308 64 10 number number NOUN ejpam-308 64 11 is be AUX ejpam-308 64 12 1 1 NUM ejpam-308 64 13 b b PROPN ejpam-308 64 14 ∑b ∑b PROPN ejpam-308 64 15 s s PROPN ejpam-308 64 16 , , PUNCT ejpam-308 64 17 t=1 t=1 ADV ejpam-308 64 18 a(b a(b PROPN ejpam-308 64 19 , , PUNCT ejpam-308 64 20 s s PROPN ejpam-308 64 21 , , PUNCT ejpam-308 64 22 t t PROPN ejpam-308 64 23 ) ) PUNCT ejpam-308 64 24 . . PUNCT ejpam-308 65 1 from from ADP ejpam-308 65 2 this this PRON ejpam-308 65 3 , , PUNCT ejpam-308 65 4 one one PRON ejpam-308 65 5 can can AUX ejpam-308 65 6 easily easily ADV ejpam-308 65 7 deduce deduce VERB ejpam-308 65 8 euler euler VERB ejpam-308 65 9 ’s ’s PART ejpam-308 65 10 congruence congruence NOUN ejpam-308 65 11 aφ(n aφ(n PUNCT ejpam-308 65 12 ) ) PUNCT ejpam-308 65 13 ≡ ≡ PROPN ejpam-308 65 14 1 1 NUM ejpam-308 65 15 mod mod PROPN ejpam-308 65 16 n n PROPN ejpam-308 65 17 for for ADP ejpam-308 65 18 ( ( PUNCT ejpam-308 65 19 a a PRON ejpam-308 65 20 , , PUNCT ejpam-308 65 21 n n CCONJ ejpam-308 65 22 ) ) PUNCT ejpam-308 65 23 = = SYM ejpam-308 66 1 1 1 X ejpam-308 66 2 . . PUNCT ejpam-308 67 1 in in ADP ejpam-308 67 2 fact fact NOUN ejpam-308 67 3 , , PUNCT ejpam-308 67 4 the the DET ejpam-308 67 5 expression expression NOUN ejpam-308 67 6 in in ADP ejpam-308 67 7 the the DET ejpam-308 67 8 lemma lemma PROPN ejpam-308 67 9 can can AUX ejpam-308 67 10 be be AUX ejpam-308 67 11 re re VERB ejpam-308 67 12 - - VERB ejpam-308 67 13 written write VERB ejpam-308 67 14 as as ADP ejpam-308 67 15 1 1 NUM ejpam-308 67 16 n n NUM ejpam-308 67 17 n∑ n∑ PROPN ejpam-308 67 18 s s PROPN ejpam-308 67 19 , , PUNCT ejpam-308 67 20 t=1 t=1 PUNCT ejpam-308 67 21 a(n a(n PROPN ejpam-308 67 22 , , PUNCT ejpam-308 67 23 s s PROPN ejpam-308 67 24 , , PUNCT ejpam-308 67 25 t t PROPN ejpam-308 67 26 ) ) PUNCT ejpam-308 67 27 = = SYM ejpam-308 68 1 ∑ ∑ PUNCT ejpam-308 68 2 d|n d|n PROPN ejpam-308 68 3 φ(n φ(n PROPN ejpam-308 68 4 / / SYM ejpam-308 68 5 d d NOUN ejpam-308 68 6 ) ) PUNCT ejpam-308 68 7 ∑ ∑ PUNCT ejpam-308 68 8 l l NOUN ejpam-308 68 9 |d |d NOUN ejpam-308 68 10 alφ(d alφ(d NOUN ejpam-308 68 11 / / SYM ejpam-308 68 12 l l NOUN ejpam-308 68 13 ) ) PUNCT ejpam-308 68 14 d d NOUN ejpam-308 68 15 . . PUNCT ejpam-308 69 1 references reference NOUN ejpam-308 69 2 15 15 NUM ejpam-308 69 3 references reference NOUN ejpam-308 69 4 [ [ X ejpam-308 69 5 1 1 NUM ejpam-308 69 6 ] ] SYM ejpam-308 69 7 i.erovenko i.erovenko PROPN ejpam-308 69 8 & & CCONJ ejpam-308 69 9 b.sury b.sury ADJ ejpam-308 69 10 , , PUNCT ejpam-308 69 11 commutativity commutativity NOUN ejpam-308 69 12 degree degree NOUN ejpam-308 69 13 of of ADP ejpam-308 69 14 wreath wreath NOUN ejpam-308 69 15 product product NOUN ejpam-308 69 16 of of ADP ejpam-308 69 17 finite finite ADJ ejpam-308 69 18 abelian abelian ADJ ejpam-308 69 19 groups group NOUN ejpam-308 69 20 , , PUNCT ejpam-308 69 21 bulletin bulletin NOUN ejpam-308 69 22 of of ADP ejpam-308 69 23 the the DET ejpam-308 69 24 australian australian ADJ ejpam-308 69 25 math math NOUN ejpam-308 69 26 . . PUNCT ejpam-308 70 1 soc soc PROPN ejpam-308 70 2 . . PUNCT ejpam-308 71 1 , , PUNCT ejpam-308 71 2 vol vol NOUN ejpam-308 71 3 . . PROPN ejpam-308 72 1 77 77 NUM ejpam-308 72 2 ( ( PUNCT ejpam-308 72 3 2008 2008 NUM ejpam-308 72 4 ) ) PUNCT ejpam-308 73 1 p.31 p.31 PROPN ejpam-308 73 2 - - PUNCT ejpam-308 73 3 36 36 NUM ejpam-308 73 4 . . PUNCT