Question

The conjugate of 1 is isomorphic to​

Answer 1

Proof:

Let G be a group and H is a subgroup of G. Suppose a is any element of G. Now ϕ:H→aHa−1 be defined by ϕ(h)=aha−1. So

ϕ(h1h2)=ah1h2a−1=ah1a−1ah2a−1=ϕ(h1)ϕ(h2)

Thus, ϕ is a homomorphism.

(one-to-one)

Now

h∈ker(ϕ)if and only ifif and only ifif and only ifϕ(h)=0aha−1=0h=0

Hence, ker(ϕ)={0}. Thus ϕ is one-to-one.

(onto)

Let y∈aHa−1. Then y=aha−1 for some h∈H. As h∈H, ϕ(h)=aha−1=y. Thus ϕ is onto.

Hence, ϕ is an isomorphism.

hope it helps

Answer 2

Answer:

to isomorphic 1 of conjugate the