Non-Abelian Groupadvanced series
This is not a puzzle so much as a way of checking that you understand the use of quantifiers.
A group is any set of things (its elements) on which is defined a binary operation * satisfying the following conditions:
Associativity: for all elements x, y and z,
(x * y) * z = x * (y * z).
Identity: there is a special element e such that for every element x,
x * e = x.
Inverse: for every element x the inverse element i(x) satisfies
x * i(x) = e
and
i(x) * x = e.
A group is said to be “abelian” if it additionally satisfies the commutative law: for all x and y,
x * y = y * x.
Find a group that is not abelian.
In your expression of this problem, do not use any constant symbol, and do not use any function symbol other than *.
Note:
For efficiency, you may as well enforce the condition that e is the smallest element (since it could always be made the smallest just by re-ordering everything). Another useful derivable fact is
e * x = x.
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