Solving the Group Theory Conundrum: Proving 'a' is in Z(G)".

[astronut24]

i've just started out with a course in group theory...here's a question that's been bothering me for a while now...
let G be a group and 'a' ,a unique element of order 2 in G. show that a belongs to Z(G).
if every element of the group has order 2 this is pretty easy...but that's not the case. one thing I've noted is a = a^-1..but does that help?

 

[Muzza]

Given any x in G, what can be said about xax^-1? For example, what is its order?

 

[astronut24]

another question...

well...thanks...the order of xax^-1 is 2...and a is the only element with order 2...so xax^-1=a and that implies the result.
another problem that i seem to be unable to figure out is...
f:(Zm , +m) --> (Zn,+n) is a group homomorphism where Zm and Zn denote groups of residue classes modulo m and n respectively. if m and n are relatively prime, then show that f is identically 0.
i fathom we are supposed to use the relation... (m,n)=1 implies am+bn =1 for some a and b in Z...how do you proceed further?
thanks for the help...

 

[matt grime]

Lemma: if f is a group homomorphism from G to H then the order of f(g) divides the order of g

Prove it and deduce the answer you want.

the am+bn=1 won't help since you aren't actually doing anything with m and n together.