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ADH wrote:
Hi all,
I'm trying to solve the following problem on groups.
a,) Let G be a group. If a belongs to G, then the order of a is the
same as the order of its inverse.
Hint: If a has order n then a^(n-1) = a^(-1), a^(-1) being the notation
for the inverse of a.
b.) Let G be a group. Let a,b belong to G. Show that a and b have the
same order if b=g^(-1)ag, for some element g belonging to G.
Hint: If b has order n then what can you say about b^n with the relation
specified? What is (g^(-1)ag)^k for any k? What happens if a has an
order less than n?
c.) Let G be a group. Show that ab and ba have the same order.
Hint: If (ab) has order n, then try to rewrite (ba)^n (to see it you
could try specific integers instead of n, say 2 or 3 to see why this works.
I believe the right and left cancellation laws will come into play in
b.) and c.).
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