Let H be a subgroup of G, let a be a fixed element of G, and let K be the set of all elements of the form a*h*a^(-1), where h E H. That is, K = {x E G | x= a*h*a^(-1) for some h E H} Prove or disprove that K is a subgroup of G
Let H be a subgroup of G, let a be a fixed element of G, and let K be the set of all elements of the form a*h*a^(-1), where h E H. That is, K = {x E G | x= a*h*a^(-1) for some h E H} Prove or disprove that K is a subgroup of G
Is necessarily of this form? Of course not!
So, think any non-abelian group and try to find a counter-example. Start with .
It is. Swlabr had a typo there. He meant , and your result should be , which shows that every element in K has an inverse.
In fact K is just a conjugate subgroup of H. You'll know more about it once you learned group action on sets and Sylow theorems.