추상대수학, 그 스물한 번째 이야기 | 군의 작용 ( Group Action )  By 초코맛 도비

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In this post, we will learn about the group action. In mathematics, a group action on a set is a group homomorphism of a given group into the group of permutations of the set. Let's look at the definition below.

 

Definition 1.

Let G be a group and let S be a set. An action of G on S is a homomorphism π:GPerm(S) of G into the group of permutations of S. We then call S a G-set. We denote the permutation associated with an element xG by πx and thus the homomorphism is denoted by xπx. We often abbreviate the notation and write simply xs instead of πx(s).

 

With the simpoler notation above, we have the two familiar properties. Look at the theorem below:

 

Theorem 1.

The following statements are satisfied:
(a) For all x,yG and sS, we have x(ys)=(xy)s.
(b) If e is the identity element of G, then es=s for all sS.

 

Proof.

Part (a)

Since xπx is a group-homomorphism, x(ys)=πx(πy(s))=(πxπy)(s)=πxy(s)=(xy)s.

 

Part (b)

Since xπx is a group-homomorphism, πe must be an identity element of the group Perm(S). Thus, πe is the identity map on S.

 

Conversely, if we are given a mapping G×SS, denoted by (x,s)xs, satisfying these two properties, then for each xG the map sxs is a permuation of S, which we then denote by πx(s). Then xπx is a homomorphism of G into Perm(S). So an action of G on S could also be defined as a mapping G×SS satisfying the above two properties. The most important examples of representations of G as a group of permutations are the following.

 

Example 1. Conjugation

For each xG, let cx:GG be the map such that cx(y)=xyx1. Then it is immediately verified that the association xcx is a homomorphism GAut(G), and so this map gives an action of G on itself, called conjugation. The kernel of the homomorphism xcx is a normal subgroup of G, which consists of all xG such that xyx1=y for all yG, i.e., all xG which commute with every element of G. This kernel is called the center of G. automorphisms of G of the form cx are called inner.

To avoid confusion about the action on the left, we don't write xy for cx(y). Sometimes, one writes cx1(y)=x1yx=yx, i.e., one uses exponential notation, so that we have the rules yxz=(yx)z and ye=y for all x,y,zG. Similarly, xy=xyx1 and z(xy)=zxy.

We note that G also acts by conjugation on the set of subsets of G. Indeed, let S be the set of subsets of G, and let AS be a subset of G. Then xAx1 is also a subset of G which may be denoted by cx(A), and one verifies trivially that the map (x,A)xAx1 of G×SS is an action of G on S. We note in addition that if A is a subgroup of G then xAx1 is also a subgroup, so that G acts on the set of subgroups by conjugation.

 

Example 2. Translation

For each xG we define the translation Tx:GG by Tx(y)=xy. Then the map (x,y)xy=Tx(y) defines an action of G on itself.

Similarly, G acts by translation on the set of subsets, for if A is a subset of G, then xA=Tx(A) is also subset of G. If H is a subgroup of G, then Tx(H)=xH is in general not a subgroup but a coset of H, and hence we see that G acts by translation on the set of cosets of H. We denote the set of left cosets of H by G/H. Thus even though H need not be normal, G/H is a G-set. It has become customary to denote the set of right cosets by GH.

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