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?
Consider G = (Z12, +). Let H = {0, 3, 6, 9}.
a. Show that H is a subgroup of G.
b. Find all the cosets of H in G and denote this set by G/H.
[Note: If x ∈ G then H +12 [x]12 = {[h +
x]12?? | [h]12 ∈ H} is the coset generated by
x.]
c. For H +12 [x]12, H +12
[y]12 ∈ G/H define
(H+12[x]12)⊕(H+12[y]12)
by(H+12 [x]12)⊕(H+12
[y]12)=H+12 [x+y]12.
d. Show that ⊕ is...
Let G = Z4XZ3XZ2 and consider the two cyclic subgroups H = h(0;
1; 1)i and K = h(2; 1; 1)i of G. (a) Find all cosets (along with
the elements they contain) to H and K, respectively. (b) Write down
Cayley tables for the factor groups G=H and G=K, and classify them
according to the Fundamental Theorem of Finite Abelian Groups.
Let
G be a finite group and H a subgroup of G. Let a be an element of G
and aH = {ah : h is an element of H} be a left coset of H. If b is
an element of G as well and the intersection of aH bH is non-empty
then aH and bH contain the same number of elements in G. Thus
conclude that the number of elements in H, o(H), divides the number
of elements...
Let G, H be groups and define the relation ∼= where G ∼= H if
there is an isomorphism ϕ : G → H.
(i) Show that the relation ∼= is an equivalence relation on the
set of all groups.
(ii) Give an example of two different groups that are
related.
Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂ G be a
subgroup of G such that K ⊂ H Suppose that H is also a normal
subgroup of G. (a) Show that H/K ⊂ G/K is a normal subgroup. (b)
Show that G/H is isomorphic to (G/K)/(H/K).
Suppose G is a group and H and H are both subgroups of G.
Let HK={hk, h∈H and k ∈K}
a.give a example such that |HK| not equal to |H| |K|
b. give a example to show f :HK →H ⨯K given by f(hk) = (h,k) may
not be well defined.
Let H and K be subgroups of a group G so that for all h in H and
k in K there is a k' in K with hk = k'h. Proposition 2.3.2 shows
that HK is a group. Show that K is a normal subgroup of HK.