Some Relations between Isologic and Varietal Perfect Groups

Copyright © 2016 S. Lotfi and S. M. 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. In 1940, Hall introduced the notion of V-isologism, with respect to a given variety of groups θ. In the present article, we study the concepts of V-perfect groups, V-subgroup, and V-quotient irreducible groups, with respect to a given variety of groups θ. Also we prove and obtain some results.


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 2.3]).Let  be a variety of groups and  be a normal subgroup of a group .Then the following statements hold:  Theorem 2 (see [7,Theorem 2.4]).Let  be a variety of groups and  be a group with a subgroup  and a normal subgroup 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 ]( 1 , . . .,   ) ∈  and all  1 , . . .,   ∈ , we have (]( 1 , . . .,   )) = ](ℎ 1 , . . ., ℎ  ), whenever ℎ  ∈ (   * ()) for  = 1, . . ., .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 Proof.One notes that Lemma 4 (ii) gives the "if" part.Now assume that  induces a -isologism between  and ; then

𝑉-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,

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][9][10][11][12].
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].The following corollary is an immediate consequence of the above theorem.
Corollary 15.Let  and  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, if  is -perfect and -perfect group, then () =   , so  is not necessarily a -perfect group.
and only if  is both -perfect and -perfect group.
Proof.(i) and (ii) can be easily obtained by the above notations and Corollary 15.

𝑉-Subgroup and 𝑉-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].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 7.2 of [7] for more information.
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 A = { |  is a normal subgroup of  with  ∩ () = 1}.The set A is nonvoid because it contains the trivial subgroup.We define a partial ordering on A by inclusion and evidently, by Zorn's Lemma, we can find a maximal normal subgroup  in A. Since  ∩ () = 1, it follows that  ∼ / by Lemma 4. Now, suppose that / is a normal subgroup of / such that (/) ∩ (/) = 1.Therefore, using the Dedekind's modular law and Proposition 1, we have  ∩ () ⊆ .Since  ∩ () = 1, we conclude that  ∈ A. 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  1 and  2 be two -isologic groups.If  1 is -subgroup and -quotient irreducible group, then so is  2 .
Now using Lemma 4, we have  = () * () and  * ()∩ () = 1.Hence  ≅  ×  * ().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  ∩ () = 1.Hence  = 1.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  * () = 1, then  ≅ . :  () →  () .
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  ∩ () = 1.