Prove that a group of even order must have an element of order 2.

Lewis Harvey 2021-01-06 Answered
Prove that a group of even order must have an element of order 2.
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joshyoung05M
Answered 2021-01-07 Author has 97 answers

To show: thereexist some x in G such that order of x is 2.
Let S={xG|x|>2}
Let lS
Since order of any element in group is equal to order of its inverse, thus |l1|=|l|>2
Hence, llS
Since, |l|>2 it implies that ll1. Hence every lement in S can be paired off with its inverse which is also on S. Thus, S otains even number of elements
GS={xG|x=eor|x=2}
Since , identity element is unique in a group, therefore G must have an element of order 2 in order to GS with even number of elements.
Therefore, a group of even order must have an element of order 2.

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