Question

In: Advanced Math

Let LaTeX: GG be an abelian group. Let LaTeX: H = { g \in G \mid...

Let LaTeX: GG be an abelian group. Let LaTeX: H = { g \in G \mid g^3 = e }H = { g ∈ G ∣ g 3 = e }. Prove or disprove: LaTeX: H \leq GH ≤ G.

Solutions

Expert Solution

Given that is an abelian group.

Then .

Also given that , being the identity element.

Theorem: a subset of a group is a subgroup of if and only if .

To prove :

Let

Then

Now

Since are also in is a subset of

Then .

Therefore we get

This is true for all .

Hence by using the above theorem we can conclude that

is a subgroup of , i.e, .

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let G be an abelian group. (a) If H = {x ∈ G| |x| is odd},...

Let G be an abelian group. (a) If H = {x ∈ G| |x| is odd}, prove that H is a subgroup of G. (b) If K = {x ∈ G| |x| = 1 or is even}, must K be a subgroup of G? (Give a proof or counterexample.)

Let G be an abelian group and K is a subset of G. if K is...

Let G be an abelian group and K is a subset of G. if K is a subgroup of G , show that G is finitely generated if and only if both K and G/K are finitely generated.

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is...

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

(a) Let G be a finite abelian group and p prime with p | | G...

(a) Let G be a finite abelian group and p prime with p | | G |. Show that there is only one p - Sylow subgroup of G. b) Find all p - Sylow subgroups of (Z2500, +)

Theorem 2.1. Cauchy’s Theorem: Abelian Case: Let G be a finite abelian group and p be...

Theorem 2.1. Cauchy’s Theorem: Abelian Case: Let G be a finite abelian group and p be a prime such that p divides the order of G then G has an element of order p. Problem 2.1. Prove this theorem.

PLEASE SHOW ALL STEPS WITH EXPLANATIONS Let G be a group (not necessarily an Abelian group)...

***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 G be a finite group and H a subgroup of G. Let a be an...

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 an abelian group and n a fixed positive integer. Prove that the following...

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 group acting on a set S, and let H be a group...

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 ⊂...

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).

Subjects