Question

1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a) any non-identity element equals its own inverse b) G is an abelian group 4- (1+1+1,5+,5 marks) Let I and J ideals in a ring R. Verify that a) I J is an ideal in R b) + is an ideal in R defined by f(a)-a + is a surjective homomorphism of rings. d) 5 (1,5+3 marks) a) Let R be a ring and e E R an element such that e2 -e, that is not a zero divisor Prove that e is the identity element of R b) Let S be a ring with identity is,Ta ring without zero divisors and f:S → Ta non- zero homomorphism of rings (at least one element in S is not mapped to 0T). Prove that i) f(is)-Or i) (f(is)) - fls) ii f(1s)-1T

Answer 1

єн. Naw e H and A 0 tence ad

12 12 12 123 2 1 2. gcd(12,3) 2. 12 Since, 5,311 one prume 40 12 , so order t e e a.

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