Question

Let G be the group Z3 + Z4 and let H = h(1, 2)i be the cyclic subgroup generated by (1, 2).

(a) Find the index [G : H] of H in G.

(b) Is H a normal subgroup of G? Justify your answer.

Answer #1

Let G be a finitely generated group, and let H be normal
subgroup of G. Prove that G/H is finitely generated

1) Let G be a group and N be a normal subgroup. Show that if G
is cyclic, then G/N is cyclic. Is the converse true?
2) What are the zero divisors of Z6?

Let G be a group and suppose H = {g5 : g ∈ G} is a
subgroup of G.
(a) Prove that H is normal subgroup of G.
(b) Prove that every element in G/H has order at most 5.

Let H=<(2 3)> be the cyclic subgroup of G=S3
generated by the transposition (2 3). Write (as sets) the
right-cosets and left-cosets of H in G

a) Let H be a subgroup of a group G satisfying [G ∶ H] = 2. If
there are elements a, b ∈ G such that ab ∈/ H, then prove that
either a ∈ H or b ∈ H. (b) List the left and right cosets of H =
{(1), (23)} in S3. Are they the same collection?

Let G be a finite group and H be a subgroup of G. Prove that if
H is
only subgroup of G of size |H|, then H is normal in G.

Let G be a finite group and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G.
Hint: Consider the subgroup aHa-1 of G.
Please explain in detail!

Let G be a finite group, and suppose that H is normal subgroup
of G.
Show that, for every g ∈ G, the order of gH in G/H must divide
the order of g in G.
What is the order of the coset [4]42 +
〈[6]42〉 in Z42/〈[6]42〉?
Find an example to show that the order of gH in G/H does not
always determine the order of g in G. That is, find an example of a
group G, and...

f H and K are subgroups of a group G, let (H,K) be the subgroup
of G generated by the elements {hkh−1k−1∣h∈H, k∈K}.
Show that :
H◃G if and only if (H,G)<H

let g be a group. let h be a subgroup of g. define
a~b. if ab^-1 is in h. prove ~ is an equivalence relation on g

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