Journal of Mathematics
Volume 2016 (2016), Article ID 5093983, 4 pages
http://dx.doi.org/10.1155/2016/5093983

Research Article

Some Relations between Isologic and Varietal Perfect Groups

Shokufeh Lotfi and S. Mostafa Taheri

Faculty of Mathematics, Golestan University, Gorgan, Iran

Received 26 July 2016; Accepted 15 November 2016

Academic Editor: Shaofang Hong

Copyright © 2016 Shokufeh Lotfi and S. Mostafa Taheri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In 1940, Hall introduced the notion of -isologism, with respect to a given variety of groups . In the present article, we study the concepts of -perfect groups, -subgroup, and -quotient irreducible groups, with respect to a given variety of groups . Also we prove and obtain some results.

1. Introduction

In 1940, Hall introduced the notion of isoclinism, which is an equivalence relation on the class of all groups such that all abelian groups fall into an equivalence class. This notion is weaker than isomorphism and plays an important role in classification of finite -groups in [1]. Later he generalized the notion of isoclinism to the notion of -isologism, with respect to a given variety of groups in [2]. If is the variety of all the trivial groups, abelian groups, or nilpotent groups of class at most , then -isologism coincides with isomorphism, isoclinism, and -isoclinism, respectively; for more information see [1, 3]. The purpose of this article is to show some properties of -perfect groups, -subgroup, and -quotient irreducible groups, with respect to a given variety of groups .

Throughout the paper, we assume that is the variety of groups defined by the set of words and the notations are taken from [4]. denotes the verbal subgroup and the marginal subgroup of with respect to ; see [5] for more information on varieties of groups.

For a group with a normal subgroup , is defined (following [6]) to be the subgroup of generated by the following set:

One may easily show that is the smallest normal subgroup of contained in , such that .

The following results give basic properties of the verbal and the marginal subgroups of a group with respect to the variety , which is useful in our investigations; see [7] for more information.

Proposition 1 (see [7, Proposition ]). Let be a variety of groups and be a normal subgroup of a group . Then the following statements hold:

(i), .(ii).(iii).(iv), .(v), in particular .(vi)If , then and .(vii)If , then , in particular .

Theorem 2 (see [7, Theorem ]). Let be a variety of groups and be a group with a subgroup and a normal subgroup . If , then .

The following definition from [3] is vital in our investigations.

Definition 3. Let be a variety of groups defined by the set of laws and let and be two groups. Then is said to be a -isologism between and , if

are isomorphisms such that, for all and all , we have , whenever for . In this case, we write and we will say that is -isologic to .

In particular, if is the variety of abelian groups we obtain the notion of isoclinism due to Hall [1].

The following lemma is needed in our investigations. For more information see Lemma of [7].

Lemma 4. Let be a variety of groups and , are a subgroup and normal subgroup of group , respectively. Then the following statements hold:

(i). In particular, if , then . Conversely, if satisfies the descending chain condition on subgroups and , then .(ii). In particular, if , then . Conversely, if satisfies the ascending chain condition on normal subgroups and , then .

Now, by Lemma 4, we obtain the following theorem.

Theorem 5. Let be a variety of groups and be a subgroup of a group . If is an epimorphism from onto , then induces a -isologism between and if and only if .

Proof. One notes that Lemma 4 (ii) gives the “if” part. Now assume that induces a -isologism between and ; then is an isomorphism. Hence .

2. -Perfect Groups

This section is devoted to study -perfect groups, which are vital in our investigations.

The following definition is essential in our further study.

Definition 6. A group is said to be -perfect with respect to the variety , if .

In particular, if is the variety of abelian groups, then -perfect groups coincide with perfect groups.

The following theorems give the connections between -perfect and -isologism groups.

Theorem 7. Let be a variety of groups and be a finite -perfect group with trivial marginal subgroup. Then any -isologic group to is isomorphic to the direct product of by the marginal subgroup of .

Proof. By the assumption,

Now using Lemma 4, we have and . Hence .

Theorem 8. Let be a variety of groups and be a finite group. If is a -perfect subgroup of such that , then .

Proof. By Lemma 4, we have and . Hence .

Theorem 9. Let be a variety of groups and be a finite -perfect group. Then can not be -isologic to any proper subgroup or factor group of itself.

Proof. If is a subgroup of such that , then, by Lemma 4, it follows that . Hence .

Theorem 10. Let be a finite group and be a group of the same order and isologic to , with respect to a given variety . If is -perfect or , then .

Proof. By the definition of isologism, we have the following isomorphisms:

Now, clearly if , then . Since , it implies that and hence . If , then the result follows immediately.

3. Product of Varieties

In 1976, Leedham-Green and Mckay [6] introduced the notion of the product of varieties as follows.

Let and be varieties of groups defined by the set of words and , respectively. The product is the variety of all groups such that . They also showed that the verbal subgroup of the product is .

In this section, using the notion of the product of varieties we present some results. Also further information about product of varieties and varietal isologism may be found in [8–12].

The following theorem gives a set of defining words for this product variety. It can be used in computations with the words. For more information see ([6, Proposition ]).

Theorem 11. Using the above notations and definitions, assume and are words, and for each

Then is defined by each of the following sets:

With respect to the marginal subgroup of a group corresponding to the variety , we have the following theorem.

Theorem 12 (see [7, Theorem ]). Let and be two varieties and put . Then for any group the following statements hold:

(i).(ii).

Leedham-Green and Mckay proved that this product featuring in is not commutative in [6]. Also, Hekster proved that this product is not associative in [7].

Now, considering the products of varieties, Neumann defined that the notions of are the variety whose set of laws are in , and consists of all groups whose -subgroups centralize -subgroups.

The following lemma gives the connection of the above product varieties, which was already proved by Hekster in [7].

Lemma 13. Let and be two varieties of groups. Then and .

Now, we are able to obtain the following theorem.

Theorem 14. Let and be two varieties of groups and . Then a groupis -perfect if and only if is -perfect and -perfect group.

Proof. Let be a -perfect group. Then by Proposition 1, we have

Therefore, is -perfect and -perfect group.
Conversely, if , then

The following corollary is an immediate consequence of the above theorem.

Corollary 15. Letand be two varieties of groups, , and be a -perfect group. Then if and only if .

Now by the virtue of the above products of varieties we have the following theorem.

Theorem 16. Let and be two varieties of groups and be an arbitrary group. Then we have the following:

(i)If and is either -perfect or -perfect group, then is not -perfect group.(ii)If and is -perfect group, then is -perfect and -perfect group.

Conversely, ifis -perfect and -perfect group, then , sois not necessarily a -perfect group.

(iii)If or , then is -perfect group if and only if is both -perfect and -perfect group.

Proof. (i) and (ii) can be easily obtained by the above notations and Corollary 15.
(iii) Let . Then . Now, assume that is -perfect group. So , which implies that is both -perfect and -perfect group. The part “only if ” is trivial.
Let . Then . Now, if is -perfect group, then, by Lemma 13, we have

Thus , and so is both -perfect and -perfect group. The part “only if ” is trivial.

4. -Subgroup and -Quotient Irreducible Groups

Hekster by the work of Stroud [13] introduced the notions of subgroup irreducible groups and quotient irreducible groups in [7].

In this final section, by using the notions of subgroup irreducible groups and quotient irreducible groups and discussion of the previous sections we give and prove our main results, which are somehow similar to those given in [7].

The following definition is introduced by Hekster in [7].

Definition 17. Let be a variety of groups. A group is called subgroup irreducible with respect to -isologism if contains no proper subgroup satisfying . A group is called quotient irreducible with respect to -isologism if contains no nontrivial normal subgroup satisfying .

In this article, we present the notions of -subgroup irreducible groups, that is, the subgroup irreducible groups with respect to -isologism, and -quotient irreducible groups, that is, the quotient irreducible groups with respect to -isologism.

The proof of the following lemma is straightforward; see Proposition of [7] for more information.

Lemma 18. Let be a variety and suppose . Then is both -subgroup and -quotient irreducible group.

A simple application of Zorn’s lemma shows that, given a group and a variety , one can always find a -quotient irreducible group. Hence this establishes the following theorem.

Theorem 19. Let be a variety of groups. If is an arbitrary group, then there exists a normal subgroup of such that and is a -quotient irreducible group.

Proof. Consider . The set is nonvoid because it contains the trivial subgroup. We define a partial ordering on by inclusion and evidently, by Zorn’s Lemma, we can find a maximal normal subgroup in . Since , it follows that by Lemma 4. Now, suppose that is a normal subgroup of such that . Therefore, using the Dedekind’s modular law and Proposition 1, we have . Since , we conclude that . On the other hand, we have and so, by the maximality of , it follows that . Therefore is trivial and hence is -quotient irreducible group.

Remark 20. Let be the product of varieties and . If is a -subgroup and -quotient irreducible group, then one notes that is -subgroup and -quotient irreducible group and also -subgroup and -quotient irreducible group.

The following theorem gives a connection between perfect groups and subgroup and quotient irreducible groups.

Theorem 21. Let be a variety of groups. If is a -perfect group, then is both -subgroup and -quotient irreducible group.

Proof. Suppose is a -perfect group with a subgroup such that . Therefore we conclude that and hence . It is easily verified that is -quotient irreducible group.

In the following theorem we show that the property of being subgroup and quotient irreducible group is closed with respect to isologism.

Theorem 22. Let be a variety of groups and and be two -isologic groups. If is -subgroup and -quotient irreducible group, then so is .

Proof. Let be a normal subgroup of such that . Then we have by Theorem of [7]. Hence and . Now, assume that

So , which implies that and hence . One can easily check that the result holds when is assumed to be -quotient irreducible group.

Competing Interests

The authors declare that they have no competing interests.

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