- #1

gottfried

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## Homework Statement

If [itex]Z[/itex]

_{2}={[0],[1]} with addition and multiplication modulo 2. How many elements does GL(2,[itex]Z[/itex]

_{2})={2 by 2 matrices with determinant 1 and entries from [itex]Z[/itex]

_{2}} have?

## The Attempt at a Solution

A[itex]\in[/itex]GL(2,[itex]Z[/itex]

_{2}) then det(A)=[1]={...,-3,-1,1,3,5,...}

Suppose A=|a b:c d| then det(a)=ad-bc. If a,b,c,d[itex]\in[/itex][itex]Z[/itex]

_{2}then a.d can equal [1] or [0].

If a.d=[1] then b.c=[0] so that det(A)=1

If a.d=[0] then b.c=[1] so that det(A)=1

If a.d=[1] then a=[1] and d=[1] meaning that b.c=[0] so b=[0] or [1] and c=[0].

From this one can see that there are 3 possible matrices with determinant 1 and a.d=[1].

The same logic shows that if a.d=[0] there are another 3 matrices with determinant 1.

Therefore GL(2,[itex]Z[/itex]

_{2}) has 6 elements.

I haven't been given a solution to this problem and I'm new to equivalence classes and [itex]Z[/itex]

_{n}so I'm not sure if have made any stupid mistakes in solving this problem?