Problem 1. Determine if the following maps are homomorphisms of groups (No reasons needed) (1). : R* R", (a)-2018a (2). : R* R*, (a)-a2018 (3). : R R, (a) 2018a (4), : R R, (a) = a2 (5). : GL(n, R) R", (A)-Det(A)10 (7). : R* R, (8). : R* R, (a) = 10. D(a)-logo(a2) where Problem 2. Find a homomorphism : C*-> C* such that Ker()-US Problem 3. Let A and B be groups. Find an isomorphism : A x B Bx A Problem 4. Determine if the following subgroups of GL(2,R) is a normal subgroup (no reasons needed) (1) H is the subgroup that consists of 2 x 2 invertible diagonal matrices (2) H is the subgroup that consists of 2 2 invertible upper triangular matrices (3) H is the subgroup that consists of 2 x 2 matrices with determinant 1 Problem 5. Let G be a finite group with GIp, where p is a prime Suppose G is not cyclic, prove that every a E G with a f e has order p