Let (G,+) be an abelian group and U a subgroup of G. Prove that
G is the direct product of U and V (where V a subgroup of G) if
only if there is a homomorphism f : G → U with f|U =
IdU
Let G be a cyclic group generated by an element a.
a) Prove that if an = e for some n ∈ Z, then G is
finite.
b) Prove that if G is an infinite cyclic group then it contains
no nontrivial finite subgroups. (Hint: use part (a))
Let G be a group. For each x ∈ G and a,b ∈ Z+
a) prove that xa+b = xaxb
b) prove that (xa)-1 = x-a
c) establish part a) for arbitrary integers a and b in Z
(positive, negative or zero)
***PLEASE SHOW ALL STEPS WITH EXPLANATIONS***
Let G be a group (not necessarily an Abelian group) of
order 425. Prove that G must have an element of order
5
Let a be an element of a finite group G. The order of a is the
least power k such that ak = e.
Find the orders of following elements in S5
a. (1 2 3 )
b. (1 3 2 4)
c. (2 3) (1 4)
d. (1 2) (3 5 4)