Question

If ( a ) is an element of a group ( G ) with ( |a|=2 m ), prove ( left|aright|=m ).

Answer 1

To prove that <|a| = m|>, use the property of exponents and the identity element of the group.

Step-by-step explanation:

To prove that
`< |a| = m| > ,` we can use the property of exponents that states
`< |(a^b)^c = a^(b*c)| > .`Since
`< |a| = 2m| > ,` we can rewrite it as
`< |a^(2m) = (a^2)^m| >`. Therefore,
`< |(a^2)^m = e^m| >` since <|a^2 = e|> where <|e|> is the identity element of the group. Using the property of exponents again, we can rewrite it as <|(a^2)^m = a^(2m) = e|>. This shows that <|(a^m)^2 = e|> which implies that <||a^m| = m|>|>.