Question Abstract Algebra?
Be G1→G2 one isomorphic in groups given x ∈ , show that O(x)= m then O(f(x)) = m
Be G one group any. Show using the Fundamental Theorem of Homomorphism that G/{e} ~ G
“~”means will isomorphic
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Suppose O(x) = m and O(f(x)) = n.
[ f(x) ]^m = f(x^m) = f(e_1) = e_2
so n <= m.
f(x^n) = [ f(x) ]^n = e_2 = f(e_1)
and since f is injective, x^n = e_1, so m <= n.
Therefore O(x) = m = n = O( f(x) ).
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To do the second proof, just use f(x) = x.
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