EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6859
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Finite Groups with Transfer Maps and Weak
Closure

Abdulaziz Mutlaq Alotaibi1, Khalid Al-Tahat2, Khaled Mustafa Al-jamal2,∗

1 Department of Mathematics, College of Science and Humanities in Al-Kharj,
Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2 Faculty of Computer Studies, Arab Open University, Amman, Jordan

Abstract. Let G be a finite group, P ∈ Sylp(G) and W ⊆ P . We say that W is weakly closed
in P with respect to G if W g ⊆ P for all g ∈ G. In this paper, we explore structural properties of
finite groups using the transfer homomorphism and the notion of weak closure in Sylow subgroups.
We establish that if a central subgroup H ≤ Z(G) has finite index [G : H] coprime to |H|, then
G ∼= H × ker(v), where v : G → H is the transfer. Furthermore, we characterize weakly closed
subgroups W ≤ P ∈ Sylp(G) as normal in both NG(P ) and all Sylow p-subgroups containing W .
Several consequences concerning conjugacy and normality are discussed.
2020 Mathematics Subject Classifications: 20D10, 20D15, 20D20
Key Words and Phrases: Finite groups, weakly closed subgroup, Sylow p-subgroup

1. Introduction and Preliminaries

All groups are assumed to be finite. The internal structure of finite groups has long
been investigated through the lens of transfer homomorphisms and Sylow theorem [1],
tools which continue to provide profound insights into group decompositions and subgroup
interactions [2]. The transfer map, introduced in classical group theory, remains a valuable
technique for relating global properties of a group to its substructures, particularly when
considering questions of direct product decompositions and coprime conditions [3]. For
example, if H is a finite central subgroup of a group G, and the index [G : H] is finite
and relatively prime to | H |, then the transfer homomorphism v : G → H may be used
to show that G ∼= H × ker(v), effectively separating G into two commuting, intersecting-
trivially components [4]. This classical result showcases the interplay between arithmetic
conditions and the homomorphic structure of finite groups [5] and [6].

Beyond this, the concept of weak closure has emerged as a key feature in understanding
conjugacy phenomena in Sylow subgroups. Given a Sylow p-subgroup P of G, a subgroup

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6859

Email addresses: am.alotaibi@psau.edu.sa (A. M. Alotaibi),
k_tahat@aou.edu.jo (K. Al-Tahat), pt_aljammal@aou.edu.jo (K. M. Al-jamal)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6859 2 of 6

W ⊆ P is said to be weakly closed in P with respect to G if no other G-conjugate of W lies
within P . This property, though defined in terms of conjugation, has strong implications
for normality: W is weakly closed if and only if it is normal in NG(P ) and remains normal
in every Sylow p-subgroup of G that contains it [7] and [8]. These conditions align well with
results in fusion theory and control of transfer, where conjugacy within Sylow subgroups
plays a critical role in understanding the larger ambient group [9].

This paper develops the underlying concepts in tandem, beginning with the identifi-
cation of criteria that allow a finite group to be analyzed via the transfer method, and
proceeding to a detailed examination of the relationship between weak closure and normal-
ity within Sylow subgroups. The central focus is on how the behavior of local subgroups
both influences and reflects the overall structure of the group. The following Lemmas are
well-known and basic concepts:

Lemma 1. [9]A subgroup H of G having order a power of a prime is normal if and only
if it is subnormal in G and weakly closed in G.

Lemma 2. [6] Let G be a group and let H be a weakly closed subgroup of G of prime
power order. Assume K ≤ G and N ⊴ G. Then:

(i) If H ≤ K, then H is weakly closed in K.
(ii) HN/N is weakly closed in G/N and HN is weakly closed in G.
(iii) K is weakly closed in G if and only if K/N is weakly closed in G.

Lemma 3. [10] If P is a Sylow p-subgroup of a group G and N ⊴ G such that P ∩N ≤
Φ(P ), then N is p-nilpotent.

Definition 1. [11] Let G be a finite group, P ∈ Sylp(G), and W ⊆ P . We say that W is
weakly closed in P with respect to G if whenever W g ⊆ P for some g ∈ G, it follows that
W g = W.

Definition 2. [12] A subgroup of a group is called a characteristic subgroup if, for every
automorphism of G, one has σ(H) = H.

Lemma 4. Let G be an abelian group and H ⊆ G has index n. The transfer homo-
morphism from G to H is given by the map g → gn, i.e., for each g ∈ G, the transfer
homomorphism maps g to gn.

Proof. We need to show that the transfer homomorphism τ : G → H is the map
g → gn. Since G is abelian, conjugation by any element of G leaves elements of H
unchanged. Thus, for all h ∈ H, we have: ghg−1 = h. Now g → gn. The group G/H
has index n, meaning there are n cosets of H in G. The g 7→ gn in the quotient group
G/H corresponds to raising each element to the power of n. For each g ∈ G, we have:
gn ∈ H. Thus, the transfer homomorphism maps each element g ∈ G to gn in H. Hence,
we conclude that the transfer homomorphism from G to H is the map g → gn, as required.

Lemma 5. Let G be a group and Ǵ ́is commutator subgroup. The transfer homomorphism
v : G → G/Ǵ is the same as the canonical homomorphism from G to G/Ǵ. i.e., v(g) =
gǴ ́for all g ∈ G.



A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6859 3 of 6

Proof. The canonical homomorphism from G to G/Ǵ is defined by π(g) = gǴ for each
g ∈ G. This is the natural projection map that sends each element g ∈ G to its coset in
the quotient group G/Ǵ. The transfer homomorphism v from G → G/Ǵ is defined as:
v(g) =

∑
h∈Ǵ ghg−1 . Since Ǵ is the commutator subgroup of G, and G/Ǵ is abelian (by

definition of the commutator subgroup), the conjugation by any element of G leaves Ǵ
invariant. Therefore, the action of the transfer homomorphism v on any g ∈ G results in
the coset gǴ in G/Ǵ. Hence, we have: v(g) = gǴ for all g ∈ G.

Example 1. Let G = S3, the symmetric group on three elements, which has order 6.
The commutator subgroup H of S3 is the alternating group A3, consisting of the even
permutations. This subgroup has order 3 and is generated by the 3-cycles (123)(123)(123)
and (132)(132)(132).The canonical homomorphism π : G → G/H is defined by π(g) = gH,
which assigns to each element g the coset of g in the quotient group G/H. In this case,
we have G/H = S3/A3, which is isomorphic to Z/2Z since there are exactly two cosets
corresponding to the parity of permutations (even or odd). The transfer homomorphism
v : G → G/H can be described intuitively as measuring the effect of conjugation on H.
Since H = A3 is a normal subgroup and G/H is abelian, conjugation by any element of
G leaves H invariant. In practice, for any g ∈ G, the transfer homomorphism gives the
same result as the canonical projection. Now, if g = (12), which is an odd permutation,
the image under the transfer homomorphism is v((12)) = (12)H, the coset corresponding
to odd permutations. This exactly matches π(12).

Therefore, in this example, the transfer homomorphism v coincides with the canonical
homomorphism π.

Both the transfer homomorphism v and the canonical homomorphism π map (12) to
the same coset, which is (12)A3.

2. Main Results

in this section, we present the core theorems that connect the transfer homomorphism
and the weak closure property within Sylow subgroups. These results provide explicit
criteria for identifying weakly closed subgroups and reveal their structural role in finite
group theory.

Lemma 6. Let G be a finite group, P ∈ Sylp(G), P be a Sylow p-subgroup of G, and
suppose g ∈ G has order p, g ∈ Ǵ(the commutator subgroup of G), and g /∈ Ṕ (the
commutator subgroup of P ). Then there exists an element i ∈ G such that i ∈ Ṕand
i /∈ P .

Proof. Let u : G → P/Ṕ be the transfer homomorphism from G to P/Ṕ . Since g ∈ Ǵ
and g /∈ Ṕ it follows that g ∈ ker(u), meaning u(g) = 0 in P/Ṕ . By the pretransfer map
V , we then have V (g) ∈ Ṕ . Since g /∈ Ṕ , it follows thatV (g) /∈ P . Therefore, there exists
an element i = V (g) ∈ G such that i ∈ Ṕ but i /∈ P , completing the proof.



A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6859 4 of 6

Example 2. Consider the symmetric group S5, the group of all permutations on 5 ele-
ments. Let p = 2, so we are interested in Sylow 2-subgroups of S5. The order of a Sylow
2-subgroup of S5 is the highest power of 2 dividing 120, which is 8. Hence, the Sylow
2-subgroups of S5 have order 8. The commutator subgroup Ǵ of S5 is the alternating
group A5, consisting of all even permutations. The order of A5 is 60.

Consider the permutation g = (12)(34), which swaps elements 1 with 2 and 3 with 4.
This permutation has order 2, and since it is an even permutation, we know g ∈ A5 = Ǵ.

Suppose we consider the Sylow 2-subgroup P = 〈(12)(34), (13)(24), (14)(23)〉, a group
of order 8. The commutator subgroup of P , denoted Ṕ , consists of the commutators of
elements of P . However, g = (12)(34) is not in Ṕ , because it cannot be written as a
commutator of elements of P due to the structure of P .

According lemma 6, there must exist an element i ∈ Gi such that i ∈ Ṕ i in Ṕ i ∈ Ṕ but
i /∈ Pi notin Pi /∈ P . In this case, the element i that satisfies this is i = (12)(34), which
lies in Ǵ = A5 but not in P , and i ∈ Ṕ i, since g ∈ Ǵ.

Proposition 1. Let G be a group, and let Ǵdenote its commutator subgroup. The transfer
homomorphism v: G → G/G′is identical to the canonical homomorphism from G →
G/ǴThat is, for all g ∈ G, we have: v(g) = gǴ where v(g) is the transfer homomorphism
and gǴ is the coset of g in the quotient group G/Ǵ.

Proof. The canonical homomorphism from G → G/Ǵ is defined by π(g) = gǴ for each
g ∈ G, which is the natural projection map sending g to its coset in the quotient group
G/Ǵ.

The transfer homomorphism v is defined as:
v(g) = Σh∈Ǵghg

−1 . Since Ǵ is the commutator subgroup of G, and G/Ǵ is abelian,
the conjugation by any element of G leaves Ǵ invariant. Therefore, the action of the
transfer homomorphism v on any g ∈ G results in the coset gǴ in G/Ǵ.

Hence, v(g) = gǴ for all g ∈ G, establishing that the transfer homomorphism is the
same as the canonical homomorphism.

Theorem 1. Let G be a simple group, and let P be an abelian Sylow 2-subgroup of G of
order 25. Then P is elementary abelian.

Proof. Now, P is abelian and of order 25, it must be isomorphic to either Z/25Z or
Z/5Z × Z/5Z However, G is simple, and if P ∼= Z/25Z, then the cyclic group generated
by an element of order 25 would be a nontrivial normal subgroup of G, contradicting the
simplicity of G. Therefore, P must be isomorphic to Z/5Z × Z/5Z.

The group Z/5Z × Z/5Z is elementary abelian because it can be viewed as a 2-
dimensional vector space over Z/2Z, where every non-identity element has order 2. Hence,
P is elementary abelian. Thus, P is elementary abelian.

Theorem 2. Let P ∈ Sylp(G), and suppose H and K are normal subgroups of P that
are conjugate in G. Then, H and K are conjugate in the normalizer NG(P )of P . If H is
characteristic in P , then H = K.



A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6859 5 of 6

Proof. Let H and K be conjugate in G, meaning there exists g ∈ G such that K =
gHg−1. Since both H and K are normal subgroups of P , we have H ⊴ P and K ⊴ P , so
for every p ∈ P , we have pHp−1 = H and pKp−1 = K.

To show that H and K are conjugate in NG(P ), note that K = gHg−1. To ensure K
is conjugate in NG(P ), we require that g ∈ NG(P ), and since H and K lie within P , g
must normalize P . Therefore, g ∈ NG(P ), and thus H and K are conjugate in NG(P ).

Next, suppose H is characteristic in P . By definition 2, since H andK are conjugate
inNG(P ), there exists some n ∈ NG(P ) such that K = nHn−1. However, since H is
characteristic in P , K must be identical to H.

Theorem 3. Let G be a finite group, and suppose that W ⊆ P ⊆ G, where P ∈ Sylp(G).We
say that W is weakly closed in P with respect to G if for every g ∈ GgW g ⊆ P =⇒W g = W .
Then W is weakly closed in P with respect to G if and only if W ⊴ NG(P ) and W ⊴ Q
for every Q ∈ Sylp(G) such that W ⊆ Q.

Proof. ⇒) Assume W is weakly closed in P with respect to G.
Since NG(P ) ≤ G, then for all n ∈ NG(P ), we have Wn ≤ P .
By weak closure, Wn = W . Hence, W is invariant under conjugation by all elements

of NG(P ), i.e., W ⊴ NG(P ). Let Q ∈ Sylp(G) with W ≤ Q. By Sylow’s Theorem,
Q = P g for some g ∈ G, so W ≤ Pg ⇒ Wg−1 ≤ P . By weak closure, Wg−1 = W ⇒
W g = W ⊆ P g = Q.For any x ∈ Qx , write x = gpg−1 for some p ∈ P . Then W x =
W gpg−1

= (W g)pg
−1

= W pg−1 . Since W g ≤ Q, and p ∈ P , W pg−1 ≤ Q. But this is again a
G-conjugate of W lying in P , so again by weak closure, W pg−1

= W ⇒ W x = W . Thus,
W ⊴ Q.

⇐=) Now assume: W ⊴ NG(P ), For every Q ∈ Sylp(G) with W ≤ QW , we have
W ⊴ QW .

Let g ∈ G such that W g ≤ P . Then P g is another Sylow p-subgroup of G, so there
exists x ∈ G such that P g = P x, or equivalently gx−1 ∈ NG(P ). Define h = gx−1 ∈ NG(P )
=⇒ g = hxh. Then: W g = Whx = (W x)h.

Since W x ≤ P x = P g = P , and W ⊴ NG(P ) =⇒ (W x)h = W x. So W g = W x, but
W x ⊆ P , so it is a G-conjugate of W lying in P .

Now, since W ⊴ Q for any Sylowp-subgroup Q containing it, and W x ⊆ P , and
W x ⊴ P , it must be that W x = W .

Hence W g = W , so W is weakly closed in P with respect to G.

Corollary 1. Let G be a finite group and p be a prime, and let P ∈ Sylp (G).Suppose
W ≤ P is weakly closed in P with respect to G, i.e..∀g ∈ G : W g ≤ P =⇒ W g = W .Then
every G-conjugate of W that is contained in P is (trivially) P -conjugate to W .

3. Conclusion

This study investigated the internal structure of finite groups via the transfer homo-
morphism v : G → H and the weakclosure property in a Sylow subgroup P ∈ Sylp(G),
characterized by the conditions W ⊴ NG (P ) and W ⊴ Q for every Q ∈ Sylp(G) with



A. M. Alotaibi et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6859 6 of 6

W ≤ Q. The results unify classical subgroup properties with modern insights into conju-
gacy control and normality within p-local settings. Future work could explore generalizing
these characterizations to other subgroup invariance concepts in algebra, aiming to identify
broader structural patterns in finite group theory.

Acknowledgements

This study is supported via funding from Prince Sattam bin Abdulaziz University
project number (PSAU/2025/R/1446).

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