Question

Step 1

In a group **G**, two element **g **and **h **are called conjugate when **h = x g x ^{-1}** for some

For an element g of a group **G,** its conjugacy class is the set of elements conjugate to it:

**{xgx−1 : x ****∈ G}.**

Part (a) The set **xHx ^{-1}** is a subgroup of

** **

**Proof: **Since H is a subgroup, we know that **H** is nonempty, and therefore **xHx ^{-1}** is nonempty.

Next,

let **x h _{1} x^{-1 }and x h_{2 }x^{-1 }**be elements of

Since H is a subgroup, we know that **h _{1}, h_{2} **

Finally, let xhx^{-1} be any element of **xHx ^{-1}** , where

Since H is a subgroup, we know that **h ^{-1} **

and therefore **(x h _{1} x^{-1})^{-1}= x h^{-1} x^{-1}** is an element of

Step 2

Part (b):

If **H **is a finite group, then each conjugacy class in **H **has size dividing **|H|**

Step 3

An isomorphism form H to xHx-1 is one-one mapping from H onto xHx-1 and preserve the group operation and **xHx-1** is homomorphism:

First prove that...

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