0
votes
2answers
44 views

Basic proof of statement in abstract algebra?

http://www.proofwiki.org/wiki/Abelian_Quotient_Group The third step (in both proofs) is something I am having trouble seeing. The theorem itself is not difficult to prove, but it is much cleaner this ...
1
vote
1answer
51 views

subgroup proof.

Prove that if $G$ is an abelian group, then $H =\{ x \in G\mid x^{2} = e \}$ is a subgroup of $G$. I did show that $H$ is close, associative, have identity and inverse element. Then my prof said I ...
0
votes
2answers
57 views

“Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$”

I have a question that says this: Classify $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8 / \langle(1,1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups. ...
2
votes
2answers
35 views

Ordered abelian groups

Consider the following axioms: 1) $\ x+(y+z)=(x+y)+z$ ; $\forall x \forall y \forall z$ 2) $\ x+0=x$ ; $\forall x$ 3) $\forall x$ $ \exists y$ such that $\ x+y=0$ 4) $ \ x+y=y+x$ ...
0
votes
3answers
563 views

G is an abelian group. Prove $G^{(n)}$ is a subgroup of G

Let G be an abelian group. Prove that $$G^{(n)} = \{g \in G | g^n = 1_G \}$$ is a subgroup of G. How do I go about doing this? I understand that $G^{(n)}$ is basically the set of all elements ...
2
votes
2answers
130 views

Prove $H \times G$ is commutative iff $H, G$ are commutative

This is another proof question I am asking about, can someone give me tips on how to answer these questions? My question says: "Let $H,G$ be arbitrary groups. Prove that $H \times G$ is commutative ...
1
vote
1answer
209 views

Computing Quotient Groups $\mathbb{Z}_4 \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$, $\mathbb{Z} \times \mathbb{Z}_{6}/ \langle (1, 2) \rangle$

Let $G/H = \mathbb{Z}_{4} \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$. I know that $|G/H|$ = 4, so $G/H \simeq \mathbb{Z}_{2} \times \mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$. Since $G/H$ has an ...