Suppose Cm is a finite group of even order Show that 7 conta

Suppose Cm is a finite group of even order. Show that (7 contains an element of order 2.

Solution

We know that for every g G, there exists a unique inverse of g denoted by g-1 G such that gg-1 = e. Let us assume that there is no a G such that a2 = e. Now, if   a2 e , then a a^-1. Let us now arrange the elements of G into pairs elements with their respective inverse elements. In all such pairs, no element can be its own inverse, otherwise, the order of this element will be 2. However, the identity element e is its own inverse. It implies that the order of G is odd , if e is the only element which is its own inverse. This is a contradiction. Hence, there must be at least one more element, say a G, such that a = a^-1 . Then a² = e and hence G has an element of order 2.

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