Thread: Abstract Algebra

I have no clue how to attack these question one. All help or hints is appreciated.

G always denotes a group

1. Let G be finite and G not equal to {e}. Show that G has an element of prime order.

2. Prove that isomorphic groups have isomorphic automorphism groups.

3. Let a, b be in G . If |a| and |b| are relatively prime (i.e. gcd (|a|, |b|)=1), then <a> intersection <b> ={e}. Prove the last statement.

Thanks!

Originally Posted by GreenandGold

1. Let G be finite and G not equal to {e}. Show that G has an element of prime order.

Let and construct . Let . The integer has a prime divisor . The element has order .

2. Prove that isomorphic groups have isomorphic automorphism groups.

Let be a group isomorphism.
Define by .
Show this is an isomorphism.

3. Let a, b be in G . If |a| and |b| are relatively prime (i.e. gcd (|a|, |b|)=1), then <a> intersection <b> ={e}. Prove the last statement.

Notice that is a subgroup of and . By Lagrange's theorem it means divides and . Therefore, .

Originally Posted by ThePerfectHacker

Let and construct . Let . The integer has a prime divisor . The element has order .


Let be a group isomorphism.
Define by .
Show this is an isomorphism.


Notice that is a subgroup of and . By Lagrange's theorem it means divides and . Therefore, .

So need to show that is a structure preserving function?

Originally Posted by GreenandGold

So need to show that is a structure preserving function?

Yes