Compatible pair of actions for two same cyclic groups of 2-power order

The nonabelian tensor product is defined for a pair of groups which act on each other provided the actions satisfying the compatibility conditions. In this paper, the compatible pairs of actions and cyclic groups of the 2-power order are considered. The aim of this study is to investigate the number of the compatible pair of actions for two same cyclic groups of 2power order with same order of actions. In attained the objectives stated, the necessary and sufficient conditions of cyclic groups of 2-power order act compatibly on each other are used. At the end of this paper, the exact number of compatible pair of actions for two same cyclic groups of 2-power order with a same order of actions is given.


Introduction
In 1984, [1] introduced the nonabelian tensor product for group extended from the concept by [2].It has its origins in the algebraic K-theory and homotopy theory.The nonabelian tensor product is defined in [1] as a group generated by the symbols g h ⊗ with relations ( )( ) ) and satisfy the compatibility conditions for all , g g G ′ ∈ and , h h H ′ ∈ .However, the paper by [3] in 1987 became starting point for research of nonabelian tensor product.They provided a list of open problems regarding nonabelian tensor product and nonabelian tensor square.From the given problems by [3], many researchers had studied group theoretical condition of nonabelian tensor product.
This research focused on the concept of compatibility since it is the conditions in determining the nonabelian tensor product.[4] determined the compatible conditions of actions for the finite cyclic group when one of action is trivial and both are trivial.Then, [5] presented the new necessary and sufficient conditions of finite cyclic groups of 2-power order which included the order of the action as one of the conditions.[6] also provide the compatible conditions of actions for finite cyclic group of order 2  p with the order of action is .p In 2015, [7] gave specific pairs of actions which compatible for some finite cyclic 2-groups but focused on actions of order two and four.Then [8] studies on the compatible pair of nontrivial action for finite cyclic 2-groups.While, [9] focused on compatible pair of nontrivial actions for cyclic groups of 3-power order.In this paper, the specific cases of the compatible pair of actions are determined which is when G H = with the actions have same order.Next, the general exact numbers of compatible pairs of nontrivial actions for two same cyclic groups of 2-power order with same order of actions are given.

The Preliminary Results
In this section, all related definitions and previous results on compatible conditions that have been done before are stated.We start with the definition of actions which given in the following definition.

Definition 2.2 Action [4]
Let G and H be groups.An action of G on H is a mapping, Φ : ) If G and H are cyclic groups, then the action of group G act on group H be required to have the property that the identity in G acts as the identity mapping on H. Thus, all elements in G act as automorphism H on H.Then, the definition of compatible action is given as follows.

Definition 2.1 Compatible Action [1]
Let G and H be groups which act on each other.These mutual action are said to be compatible with each other and with the actions of G and H on themselves by conjugation if By Definition 2.1, if one of actions is trivial, then any actions of H act on G are compatible.Next, the result on action of G acts trivially on H is given in the following corollary.

Corollary 2.3 [4]
Let G and H be groups.Furthermore, let G act trivially on H.If G is abelian, then for any action of H on G the mutual actions are compatible.
Next, the presentation of all automorphisms for cyclic groups of 2-power order is given in the following theorem.
, where ( ) [5] provided the necessary and sufficient conditions for a compatible pair of actions for two cyclic groups of 2-power order to act compatibly on each other as stated in Theorem 2.5 and Theorem 2.6.The compatible pair of actions when one of the actions has order two are provided in the following theorem.
[8] presented the necessary and sufficient conditions of two cyclic groups of 2-power order that actions compatible on each other when G H  as given in the following proposition.
In the next section, the total number of the compatible pairs of actions with same order for two same cyclic groups of 2-power order is determined.

Results
The necessary and sufficient conditions of two cyclic groups of 2-power order given in the previous section are used to find the number of compatible pairs of actions.The general exact numbers of compatible pairs of nontrivial actions for two same cyclic groups of 2-power order with 2 k     are determined.The following proposition shows the number of compatible pair of actions when Proof Let There is one automorphism of order one for each σ and σ ′ .Thus, there is one compatible pair of actions only.ii.
Let 1 k = .There are three actions have order two and by Proposition 2.5 and Proposition 2.7, all actions are always compatible with both actions have order two.Thus, there are nine compatible pair of actions.iii. Let Conf.Series: Journal of Physics: Conf.Series 890 (2017) 012120 doi :10.1088/1742-6596/890/1/012120

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then the number of compatible pair of actions is one.ii.If 1 k = , then the number of compatible pair of actions are nine.iii.If 2 k ≥ , then the number of compatible pair of actions are 2 k .1234567890 ICoAIMS 2017 IOP Publishing IOP Conf.Series: Journal of Physics: Conf.Series 890 (2017) 012120 doi :10.1088/1742-6596/890/1/012120 Furthermore, let   Aut(G) and    Aut(Proposition 3.1(i), only one compatible pair of actions exist when 0 k = .ii.By Proposition 3.1(ii), there are nine compatible pair of actions when 1 k = .iii.By Lemma 3.2, the number of compatible pair of actions are 2 2 4 m− − when 2 k ≥ , 2 k ≥ .By Theorem 2.4, there exist 2 k automorphisms of order 2 k and ( , ) ≥ .Furthermore, let   Aut(G) and   Aut(H) with