Question

In: Advanced Math

Let f be a group homomorphism from a group G to a group H If the...

Let f be a group homomorphism from a group G to a group H

If the order of g equals the order of f(g) for every g in G must f be one to one.

Solutions

Expert Solution

be a group homomorphism .

Suppose be two elements . Now ,

  

, where is the identity elemnt of H .

, As f is a homomorphism .

As given order of equals to so order of is equals to order of

Order of = Order of

Now as order of the identity element is always 1 so ,

Order of is 1 .

As only identity can have order 1 so ,

  

So implies .

Hence is one-to-one.

.

.

If you have any doubt or need more clarification at any step please comment.

Colby Messinger answered 1 year ago
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