Prove that a finite group is abelian if and only if its group table is a symmetric matrix.
If
is a finite group, there exists a bijective indexing map
for some natural number
; we can then define the group table
of
induced by
to be a matrix with
.
Suppose
is a finite abelian group. Then for all
we have
. Hence
is symmetric.
Suppose
is a symmetric matrix. Then for all
we have
, so that
is abelian.
If
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