Cyclic groups and order of an element

13.a) If G={g1,g1,....,gr} is an abelian group, show that g1,g2....gr equals the product of the elements of order 2. b) Prove Wilson's Theorem: If p is a prime then (p-1)! R (-1)(modp) note: R is a equivalence relation

Groups

let m be the smallest positive integer such that @^m=E for all @eS_n. Show that m=lcm(2,3,4,5,...,n). note: e denotes element of

Subgroup question

If X is a nonempty subset of a group G, let ={x1^(k1),x2^(k2)...xm^(km)|m>=1, xiEX and kiEZ for each i}. a) show that is a subgroup of G that contains x. b) show that C=H for every subgroup H such that XC=H. Thus is the smallest subgroup of G that contains X, and is called the subgroup generated by X. note: E denotes element of, and C= denotes set containment

subgroups

If K is a subgroup of H and H is a subgroup of G, is K a subgroup of G? Please justify your answer. thanks.

cyclic groups

Show that every cyclic group Cn of order n is abelian. (Moreover, show that if G is a group, so is GxG)