hi, i am very new to group theory .... just started reading infact. I could really appreciate some guidance on the following problem with which i am stuck. Let G be the set of all 2x2 matrices [a b ; c d] (ad-bc != 0) where a,b,c,d are all integers modulo p ( a prime number). Multiplication operation is defined as in the case of Matrices with the understanding that addition and multiplication of entries will be modulo p. (i) To verify that G (with the operation defined) forms a finite non-abelian group (ii) Given a prime p, to find the order of group G. I have been able to verify without much difficulty, the following : *) Closure under multiplication *) Existence of identity element *) Associativity under product With reference to the existence of inverse, i am not able to prove that the inverse element is actually part of the set. Am i missing some observation/fact ? That the group is finite and non-abelian is easy enough once the inverse thing is shown. The second part of the question requires to separate those elements from the p^4 matrices which do not satisfy the |A| != 0 condition. Thank you in advance !!