On a New Criterion for the Solvability of Non-Simple Finite Groups and m-Abelian Solvability

An important problem in the theory of groups came to light after Galois’ work [4]. *is problem is concerned with determining whether a group G is solvable or not. According to the literature, many conditions and criteria were introduced to deal with this problem. Feit and *ompson had proved that each finite group of odd order is solvable (see [4, 5]). Arad and Ward had proved Hall’s Conjecture about solvability in [6]. In [7], Dolfi et al. introduced the following criterion. G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x ∈ C and y ∈ Dsuch that x and y generate a solvable group. *rough this paper, we use the concept of (m-power closed group) [3] to introduce a sufficient condition that implies the solvability of a finite group. On the other hand, we define two new solvability systems (m-abelian/m-cyclic solvability) based on m-power closed groups. *ese notions will provide us an interesting connection between prime divisors of the order |G| of a finite group G and its solvability. In fact, m-abelian solvability turns out to be equivalent to the classical solvability whenever m is a prime divisor of |G|. We recall some basic definitions and theorems. All groups throughout this paper are considered finite.


Introduction
An important problem in the theory of groups came to light after Galois' work [4].
is problem is concerned with determining whether a group G is solvable or not. According to the literature, many conditions and criteria were introduced to deal with this problem. Feit and ompson had proved that each finite group of odd order is solvable (see [4,5]).
Arad and Ward had proved Hall's Conjecture about solvability in [6].
In [7], Dolfi et al. introduced the following criterion. G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x ∈ C and y ∈ Dsuch that x and y generate a solvable group.
rough this paper, we use the concept of (m-power closed group) [3] to introduce a sufficient condition that implies the solvability of a finite group.
On the other hand, we define two new solvability systems (m-abelian/m-cyclic solvability) based on m-power closed groups. ese notions will provide us an interesting connection between prime divisors of the order |G| of a finite group G and its solvability. In fact, m-abelian solvability turns out to be equivalent to the classical solvability whenever m is a prime divisor of |G|. We recall some basic definitions and theorems.
All groups throughout this paper are considered finite.
Definition 1 (see [1]) (a) Let G be a group and m be a fixed integer. We say that G is of exponent type m if for any x, y ∈ G, there exists z ∈ G such that x m y m � z m . (b) Let G be a group and H►G, and we say that H is m-normal in G if for any x, y ∈ G, there exists z ∈ G such that z m x m y m ∈ H. If this is the case, we denote H ► m G.
Theorem 1 (see [1] Definition 2 (see [1]). Suppose that G is a finite group of exponent type m. We say G is m-abelian if G m is abelian.

Theorem 3. Let G be an m-abelian group. en, the homomorphic image of G is also m-abelian.
Proof. It is clear by eorem 1 and the fact that the homomorphic image of an abelian group is abelian. □ Definition 3. Let G be a group. e m-th commutator of x, y∈ G is defined as e m-derived subgroup of G denoted by (G) m ′ is defined to be the subgroup generated by all m-th commutators of G.

Lemma 1.
Let G be a group of exponent type m. en,

m-Abelian Solvable Groups
Definition 4 (b) We say that G is m-abelian solvable if it has an m-abelian solvable series.

Lemma 3. Let G be a group, and we have (a) If G is (m-abelian), then it is (m-abelian solvable
e other inclusion can be proved by the same. Suppose that G is (m-abelian solvable); then, it has an (m-abelian solvable) series e . It is easy to show thatφ(H i−1 )▶ m φ(H i ), and thus we obtain an (m-abelian solvable) series e

(a) e direct product of two (m-abelian solvable) groups is (m-abelian solvable). (b) e direct product of finite number of (m-abelian solvable) groups is (m-abelian solvable).
Proof (a) Let G, H be two (m-abelian solvable) groups, and we have the following (m-abelian solvable) series: e 1 � H 0 ≤ H 1 ≤ · · · ≤ H n � H and e 2 � K 0 ≤ K 1 ≤ · · · ≤ K m � G; without affecting the generality, we can assume that n ≥ m; let the series ( * ) be

Proof.
ere is an (m-abelian solvable) series e { } � H 0 ≤ H 1 ≤ · · · ≤ H n � G, and we have that H 1 /H 0 � H 1 is (m-abelian), so H 1 is solvable and H 2 /H 1 is (mabelian); thus, it is solvable, and H 2 is solvable. By the same argument, we find that G is solvable.

Lemma 4. Let G be a group, and we have (a) If G is cyclic, then it is (m-cyclic) for each integer m. (b) If G is an (m-cyclic) group, then the homomorphic image of G is (m-cyclic). (c) If G is an (m-cyclic) group with a prime m/|G|, then G is solvable.
Proof (a) A subgroup of cyclic group is cyclic, so it is clear. (b) It is known that the homomorphic image of any cyclic group is cyclic and by eorem 1, the proof is complete. (c) It holds easily, since each m-cyclic group is m-abelian group.

Theorem 8. Let G be a group, and we have (a) If G is (m-cyclic), then it is (m-cyclic solvable). (b) If G is polycyclic, then G is (m-cyclic solvable) for each integer m. (c) e homomorphic image of any (m-cyclic solvable) group is (m-cyclic solvable).
Proof (a) If G is (m-cyclic), then it has an (m-cyclic solvable) series e { } ≤ G. (b) If G is polycyclic, then it has a subnormal series e { } � H 0 ≤ H 1 ≤ · · · ≤ H n � G with cyclic factors. Since every cyclic group is (m-cyclic) for each integer m, G must be (m-cyclic solvable). (c) Assume that G is (m-cyclic solvable); then, it has an (m-cyclic solvable) series e { } � H 0 ≤ H 1 ≤ · · · ≤ H n � G; suppose that H is a normal subgroup of G, and let is example is devoted to clarify the validity of our criterion in eorem 5.
Consider G � S 3 , the symmetric group of order 6. G is a 2 − group, since G 2 � Z 3 . e only normal subgroup of G is H � Z 3 which is a 3group since it is abelian, and thus G is solvable according to eorem 11.

Conclusion
In this article, we have introduced the concept of (m-abelian solvability) and (m-cyclic solvability) as two new generalizations of classical solvability and polycyclicity, respectively. We have discussed some elementary properties of these concepts and proved the main result through this paper which ensures that m-abelian solvability is equivalent of solvability in finite groups if m is a prime number that divides the order of the group.
is result shows a kind of connection between primes and solvability in finite groups. An interesting question came to light according to this work. is question can be asked as follows: If G is an infinite m-abelian solvable group for a prime m, then is G solvable? Journal of Mathematics Also, we have introduced a new sufficient condition for the solvability of finite non-simple group G based on m-power closed groups concept.
As a future research direction, m-abelian solvability can be extended to AH-subgroups defined in [8] and neutrosophic groups in [9].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.