Find all possible orders of elements in the group Z4 × Z5 × Z10.
For each possible order, give an example of an element of that
order, and prove that no other orders are possible.
Let G = Z4 × Z4, H = ⟨([2]4, [3]4)⟩.
(a) Find a,b,c,d∈G so that G is the disjoint union of the 4
cosets a+H,b+
H, c + H, d + H. List the elements of each coset.
(b) Is G/H cyclic?
If possible, find
A + B, A −
B, 2A, and 2A
− 5B.
(If not possible, enter IMPOSSIBLE.)
A =
3
−1
1
2
4
3
, B =
−1
1
−5
−4
3
−2
(a) A + B
(b) A − B
(c) 2A
(d) 2A − 5B