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## Main Question or Discussion Point

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 !!

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 !!