Solution: Given
Now,
a).
Now , the composition tables for are respectively below
From above composition tables it is clear that
are closed.
are associative.
all have identity element .
and inverse of each element is itself.
also all elements in composition tables commute to each other.
Thus, are abelian groups with 2 elements.
Hence, are finite abelian groups of order .
b).
Now, the composition table for is below
From above composition tables it is clear that
is closed.
is associative.
has identity element .
and inverse of each element is itself.
also all elements in composition tables commute to each other.
Thus, are abelian groups with 4 elements.
Hence, are finite abelian groups of order .
Which is the required proof.
This complete the solution.
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