All the best..
4. Suppose G is a group of order n < 0. Show that if G contains a group element of order n, then G is cyclic.
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
(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.
Let G be a finite group, and let H be a subgroup of order n. Suppose that H is the only subgroup of order n. Show that H is normal in G. [consider the subgroup of G] aha а
Use induction to show that 6 divides n3 −n whenever n is a nonnegative integer.
2. Let G {g, g. . . , gn-le} be a cyclic group of order n, H a group, and h є H. Define a function φ : G → H by φ(gi-hi for all 0 < i n-1. Show that φ is a group homomor- phism if and only if o(h) divides o(g). Warning: mind your modular arithmetic! [10]
3. Suppose R is a PID and M is a cyclic R-module of 'order' r E R, ie., M RI (r). Show that if N is a submodule of M then N is cyclic of order s for some s \Ir Conversely, if s | r show that M has a cyclic submodule of order s. 3. Suppose R is a PID and M is a cyclic R-module of 'order' r E R, ie., M RI (r). Show that if...
Suppose that G is a group of order 80. (a) Show that G is not simple. (b) How that G is solvable.
14. Suppose that G is a group in which {1} and G are the only subgroups. Show that G is finite and, in fact, is cyclic of order 1 or a prime.
b) Use a mathematical induction to show that: п 2" divides (n + 1) (n + 2) ... (2n – 1) (2n), for n = 0 , 1, 2, ... c) Prove by contradiction: If |x|< ɛ for all ɛ>0, then x = 0.