Answer to Question #47365 in Abstract Algebra for ayush kumar

Question

(a) Prove by the method of induction that 12 +22 +32+ ... + n2 n(n+1)(2n+1)/6 for n greater than equal to 1.

(b) Suppose (phie) is a homomorphism from Z30 to Z30 and Ker(phie) = {0, 10, 20}. If (phie)(23) = 9, determine all the elements that image 9 under (phie).

(c) Let G = {[ 1 0] , [ -1 0 ] , [ 1 0 ] , [-1 0 ] }
0 1 0 1 0 1 0 -1
Write down the table of operation where
the operation is matrix multiplication. Is G
a group ? Is G cyclic ? Justify your answer.