Argue that in any group, no element can have two distinct inverse elements. Be s

Question

Argue that in any group, no element can have two distinct inverse elements. Be sure you point out where you use each property of a group in this argument.

Argue that in any group, no element can have two distinct inverse elements. Be sure you point out where you use each property of a group in this argument. Let r1 and a2 be two distinct inverses of an element, say r in a group, G. We show that they must be the same. Let i be the identity element of the group, G 1 i D ar1 (inverse definition) i ED a2 (inverse definition) T1 (identity) i (22 ED z) ED ar1 (identity) (r r1) associativity) i (identity)

Explanation / Answer

Consider matrix G = {(a a b b) | a, b belongs to Q+ } with matrix binary multiplication opearation '.',
where Q+ denotes the +ve rational numbers. We define e = { 1 1 0 0 } . Also for M = { a a b b } ,
we define y(M) = 1(a+b)e. We can check that M x e = M and y(M) x M = e.
So in Group G , there is no element can have two distinct inverse elements.

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