Let G be a finite group. Let T be an element of Aut(G) such that
$T^{2} = I$ , and $x T = x \Leftrightarrow x = e$ .
Then, G is abelian.

Question

Let G be a finite group. Let T be an element of Aut(G) such thatT2=I, and xT=x⇔x=e.Then, G is abelian.

Frain4i62 · Accepted Answer

Step 1My solution of this problem is the following:Let&amp;nbsp;ϕ&amp;nbsp;be a mapping&amp;nbsp;ϕ:Gnix↦x−1(xT)∈GIf&amp;nbsp;ϕ(x)=ϕ(y), then&amp;nbsp;x−1(xT)=y−1(yT)Then,&amp;nbsp;(xT)(y−1T)=xy−1Then,&amp;nbsp;(xy−1)T=xy−1By the assumption of Problem 10,&amp;nbsp;xy−1=eSo,&amp;nbsp;x=ySo,&amp;nbsp;ϕ&amp;nbsp;is injective.Since G is a finite set,&amp;nbsp;ϕ&amp;nbsp;is also surjective.So, for every&amp;nbsp;g∈G, there exists&amp;nbsp;x∈G&amp;nbsp;such that&amp;nbsp;g=x−1(xT)Step 2Let g be an arbitrary element of G.Then, by Problem 10, there exists&amp;nbsp;x∈G&amp;nbsp;such that&amp;nbsp;g=x−1(xT)Then,gT=(x−1(xT))T=(x−1T)(xT2)=(xT)−1(xI)=(xT)−1x=(x−1(xT))−1=g−1So,&amp;nbsp;gT=g−1&amp;nbsp;holds for every&amp;nbsp;g∈GLet a, b be arbitrary two elements of G.Then,&amp;nbsp;(ab)−1=(ab)T=(aT)(bT)=a−1b−1=(ba)−1So,&amp;nbsp;ab=ba

Let G be a finite group. Let T

Answered question

Answer & Explanation

New Questions in Abstract algebra