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May
7
comment $D_8$ as a derived subgroup
Related: math.stackexchange.com/questions/361582/… and math.stackexchange.com/questions/365679/…
May
6
awarded  Caucus
May
4
answered Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.
May
4
comment Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.
@AlexanderGruber: How about this? Let $A$ be an abelian group such that for any abelian group $G$ and subgroup $H \leq G$, any homomorphism $f: H \rightarrow A$ can be extended to a homomorphism $g: G \rightarrow A$. Now consider the identity map $f: A \rightarrow A$. We can embed $A$ (any abelian group, in fact) into a divisible group $D$, so we can extend $f$ to a surjective homomorphism $g: D \rightarrow A$. Quotients of divisible groups are divisible, so $A$ is divisible. Thus for abelian groups, this type of homomorphism extension is possible if and only if the target group is divisible.
May
4
comment Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.
I think this is true, and even more generally when $\langle a \rangle$ is replaced by any subgroup of $G$ and $\mathbb{T}$ is replaced by any divisible abelian group.
May
4
comment Normal subgroup of a normal subgroup
Sure, but it's nice to see a general example.
May
4
answered Normal subgroup of a normal subgroup
May
4
comment Normal subgroup of a normal subgroup
@user75908: $G$ contains every permutation with the cycle structure $(ab)(cd)$ and conjugation by a permutation preserves cycle structure.
May
3
answered Terminology for infinite groups, all of whose subgroup have finite index.
May
2
comment Terminology for infinite groups, all of whose subgroup have finite index.
I think the only such group is $\mathbb{Z}$. If every nontrivial subgroup of $G$ is of finite index and $G$ is infinite, then $G$ is torsionfree and contains a subgroup of finite index isomorphic to $\mathbb{Z}$. The only group with this property is $\mathbb{Z}$.
Apr
30
comment Proper subgroup of $\mathbb{Q}^{+}$ with finite index
@PeterTamaroff: It was before the edit, when this question contained five different questions, and the only relation between them was group theory. I don't think questions like that belong here. Now the question is okay, but it seems that removing close votes is not possible
Apr
30
answered $\bigcup_{x \in G} xHx^{-1} \neq G$
Apr
30
answered There is an automorphism of $\mathbb Z_6$ which is not an inner automorphism
Apr
27
comment Reference of a Theorem in Group Theory
@PeterTamaroff: Something like "A transitive group $G$ on $m$ letters $a, b, c, \ldots$ contains necessarily a permutation that moves every letter"
Apr
27
answered Reference of a Theorem in Group Theory
Apr
26
comment When is the group of units in $\mathbb{Z}_n$ cyclic?
Groups are always nonempty since they contain the identity.
Apr
26
revised If N is a normal subgroup of G,decide whether np(N) | np(G) or np(G/N) | np(G)?
edited body
Apr
26
revised When is the group of units in $\mathbb{Z}_n$ cyclic?
added 697 characters in body
Apr
26
answered When is the group of units in $\mathbb{Z}_n$ cyclic?
Apr
26
answered If N is a normal subgroup of G,decide whether np(N) | np(G) or np(G/N) | np(G)?