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A Group \((G,*)\) (must (contain an identity, that is, \(\exists e \in G\)…
A Group \((G,*)\)
must
be associative, that is, if \(a, b, c \in G\) then \((a*b) * c = a * (b * c)\)
contain an identity, that is, \(\exists e \in G\) s.t \(e * a = a * e = a\) for each \(a \in G\)
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contain an inverse for each element, that is, if \(a \in G\) then \(\exists a^\prime \in G\) s.t \(a * a^\prime = a^\prime * a = e\)
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consists of
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and a binary operator, \(*\)
might be
Abelian or commutative, that is, if \(a, b \in G\) then \(a * b = b * a\)
-
non-Abelian
e.g.
\((\{\text{bijections on a set } S\}, \circ)\)
-
allows
cancellation, that is, for \(a, b, c \in G\) then if \(a * b = a * c\) or \(b * a = c * a\) then \(b = c\) # #