# Group theory

#### 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?

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#### 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...