Abstract algebra
A. Assume G is an abelian group. Let n > 0 be an integer. Prove that f(x) = ?" is a homomorphism from Got G. B. Assume G is an abelian group. Prove that f(x) = 2-1 is a homomorphism from Got G. C. For the (non-abelian) group S3, is f(x) = --! a homomorphism? Why?
let G be a finite group, prove that for every a in G there exists a positive integer n such that an=e, the identity.
(a) Let G be a cyclic group of order n. Prove that fo every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70.
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70 -
(1) Let G be a group and a € G. Prove that (a-1)" = (a")-1, Vn e N. Proof. Proof goes here.
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G.
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor dofn there is a subgroup of Ghaving order d. (b) Characterize all factor groups of Z70.
(9) Let G be a group, and let x E G have finite order n. Let k and l be integers. Prove that xk = xl if and only if n divides l_ k.
5. Let N be a normal subgroup of a group G and G/N be the quotient group of all right cosets of N in G. Prove each of the following: (a) (2 pts) If G is cyclic, then so is G/N. (b) (3 pts) G/N is Abelian if and only if aba-16-? E N Va, b E G. (c) (3 pts) If G is a finite group, then o(Na) in G/N is a divisor of (a) VA EG.
Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2 Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2