Let G be a group and a be an element of G. Let φ:Z→G be a map
defined by φ(n) =a^{n} for all n∈Z. a)Show that φ is a group
homomorphism. b) Find the image ofφ, i.e.φ(Z), and prove that it is
a subgroup ofG.
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)
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 and let N ≤ G be a normal subgroup.
(i) Define the factor group G/N and show that G/N is a
group.
(ii) Let G = S4, N = K4 = h(1, 2)(3, 4),(1, 3)(2, 4)i ≤ S4. Show
that N is a normal subgroup of G and write out the set of cosets
G/N.
Let G be an abelian group and n a fixed positive integer. Prove
that the following sets are subgroups of G.
(a) P(G, n) = {gn | g ∈ G}.
(b) T(G, n) = {g ∈ G | gn = 1}.
(c) Compute P(G, 2) and T(G, 2) if G = C8 ×
C2.
(d) Prove that T(G, 2) is not a subgroup of G = Dn
for n ≥ 3 (i.e the statement above is false when G is...
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 be a group acting on a set S, and let H be a group acting
on a set T. The product group G × H acts on the disjoint union S ∪
T as follows. For all g ∈ G, h ∈ H,
s ∈ S and t ∈ T,
(g, h) · s = g · s, (g, h) · t = h · t.
(a) Consider the groups G = C4, H = C5,
each acting...
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).