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Pass on!

I've been struggling with a problem for a while (belong while) now. It doesn't seem like it has a straight forward solution or an objection to the way I set it up. A group is a set that is endowed with a binary operation and that follows the following rules.

  1. a x b belongs to G.
  2. a x( b x c ) = (a x b) x c.
  3. There exists 0 | a x 0 = 0 x a = a.
  4. For every a, there exists a_1 | a x a_1 = a_1 x a = 0. ❗In any case, there is no rules that the operation "x" has to follow. The meaning of "x" in the context of "a x b" and that of "c x d" can be different. Keeping this in mind, can an element in a "group" have way more than just one inverse element?

Note: ( a, b, c, d belong to the group G. "x" is the binary operation ).

Raghav T S
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    Please use mathjax, as per site rules. Questions are for everyone here--not just the person asking--and this is hard to read. – Mike Jun 30 '25 at 18:57
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    What do you mean when you say that "x" can mean different things? – Andrei Jun 30 '25 at 19:07

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