# (4) Consider the 2 x 2 matrices 1-69). -:- (-1). «-6 -1) :-(1) (a) Prove that... (4) Consider the 2 x 2 matrices 1-69). -:- (-1). «-6 -1) :-(1) (a) Prove that {1,-1}, {1,a}, and {1,3} are finite abelian groups of order 2. (b) Prove that {1,-1,a,ß} is a finite abelian group of order 4, and compute the multiplication table for this group.

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.

##### Add Answer of: (4) Consider the 2 x 2 matrices 1-69). -:- (-1). «-6 -1) :-(1) (a) Prove that...
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