3
votes
3answers
149 views

A persisting element in all subgroups.

Let $G$ be a finitely generated abelian group and $a$ be a nontrival element of $G$ contained in all nontrivial subgroups of $G$. Is $G$ necessarily cyclic?
0
votes
1answer
29 views

Homomorphic image of intersection equals intersection of homomorphic images?

This is a tangent of this question. I wanted to remark in my answer that it is not generally true that given a group homomorphism $f:G\rightarrow H$ and two subgroups $X,Y\leq G$ that $$f(X\cap ...
2
votes
0answers
29 views

Separability of conjugacy classes in conjugacy separable semidirect products.

We say that group $G$ is conjugacy separable if for every $g \in G$ the set $g^G = \{cgc^{-1} \mid c \in G\}$ is closed in the profinite topology on $G$, i.e. for every $f \in G \setminus g^G$ there ...
3
votes
1answer
49 views

Semidirect products as (amalgamated) free product

It is well known that $\mathbb{Z}\rtimes \mathbb{Z}_2$ is free product $\mathbb{Z}_2 \star \mathbb{Z}_2$. Are there more examples of these kind?
3
votes
2answers
117 views

Is an arbitrary group generated by a traversal of the conjugacy classes?

Let $G$ be a group, and let $\mathcal C$ be the collection of conjugacy classes of $G$. Let $S$ be a traversal of $\mathcal C$ (that is $S$ contains exactly one element from each set in $\cal C$). ...
3
votes
1answer
40 views

Free cyclic subgroups in a non-abelian group

Is there any non-abelian group $G$ such that for each $a\in G$ and any automorphism $g:\left<a\right>\to \left<a\right>$ the function $$f:G\to G$$ $$f(x) = \begin{cases} g(x) & \text{ ...
1
vote
1answer
26 views

If the quotient by the $i$th center is cyclic, does it follow that the original group is abelian?

Let $G$ be a group such that there exists an $i$ such that $G/Z^i(G)$ is cyclic. Does it follow that $G$ is abelian? This question is a generalization of the well known fact that if $G/Z(G)$ is ...
9
votes
1answer
128 views

Are elements commutative when subgroups are?

Suppose G is a group and for every $H_1,H_2\le G$, $H_1H_2=H_2H_1$. Is $G$ abelian?
3
votes
2answers
54 views

Groups reluctant to have infinite subgroup

Is there a group with only one infinite subgroup‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
6
votes
1answer
49 views

Definition of the normalizer of a subgroup

Let $G$ be a group and $H$ a subgroup of $G$. Is there any counterexample to the assertion $N_G(H):=\{g\in G\mid gHg^{-1}=H\}=\{g\in G\mid gHg^{-1}\subset H\}$? Thanks!
3
votes
0answers
77 views

A sequence of subsets of an infinite group.

Is there an infinite group $G$ such there is not any sequence $(A_n)$ of its subsets such that always $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ?
4
votes
2answers
71 views

Decomposing an element into product of elements of finite order

If $G$ is a group and $g, h\in G$, it is possible that $g$ and $h$ have finite order, yet $gh$ has infinite order. For example, in Algebra: Chapter 0 by Paolo Aluffi, Exercise 1.12, the following is ...
4
votes
0answers
54 views

Does there exist an infinite solvable group with no normal abelian subgroups?

This is impossible if $G$ is finite and solvable, because then $G$ has a (nontrivial) minimal normal subgroup $A$, which can be shown (using a trick) to be abelian. I'm trying to mimic the same ...
1
vote
1answer
38 views

A group with bounded element orders and its minimal and maximal subgroups.

Let $n>1$ be an integer. Is there an abelian group $G$ with all elements of order less than $n$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains ...
0
votes
0answers
18 views

maximal and minimal subgroups of torsion abelian groups

Is there a torsion abelian group $G$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains a minimal (non-trivial) subgroup of $G$. 2) every proper ...
2
votes
1answer
26 views

Minimal normal subgroups in a non-torsion group

Is there a group $G$ with an element with infinite order such that every non-trivial $N \unlhd G$ contains a minimal (non-trivial) normal subgroup of $G$?
3
votes
1answer
110 views

Infinite abelian group counterexample [duplicate]

A finite group $G$ is abelian iff all its irreducible representation $\rho$ have dimension 1. I'm looking for a counter-example when $G$ is an infinite group. Are there any? EDIT We're dealing ...
0
votes
1answer
33 views

An intersection inequality in groups

Do you have an example of a group $G$ and decreasing sequences $(A_n),~(B_n)$ of its subsets such that $$\big(\bigcap_{i\in \Bbb N}A_i \big)\big( \bigcap_{j\in \Bbb N}B_j\big)\ne \bigcap_{i\in \Bbb ...
2
votes
1answer
50 views

Is the free group of rank 2 a subgroup of an infinite product of finite groups?

I want to find an example of a non-amenable infinite product on amenable groups. My idea is to show that the free group of rank 2 is a subgroup of an infinite product of finite groups. But I'm not ...
2
votes
1answer
52 views

Examples of non-finitely presented groups

I know several constructions leading to finitely generated non-finitely presented groups, using amalgamated products: Property: Let $A,B$ be two finitely presented groups. Then $A ...
2
votes
3answers
96 views

A sequence of subgroups tending to the trivial subgroup

Do you have an example of an abelian group $G$ with a sequence of mutually distinct nontrivial subgroups $(A_n)$ such that $$\dots \le A_n\le\dots \le A_2\le A_1\le A_0=G$$ and ...
5
votes
2answers
60 views

Is every finite weak group a group

Definition: Let $W$ be a set and $\circ:W\times W\rightarrow W$ be a function. We say that $(W,\circ)$ is a a weak group iff there exists unique $e\in W$ such that $\forall x\in W[x\circ e=e\circ ...
0
votes
1answer
38 views

Are there finite “similar” non-isomorphic groups

Let $G_1,G_2$ be two groups.We say that $G_1,G_2$ are similar iff for every integers $a_1,a_2,...,a_n\in \{1,-1\}$ and every function $f:\{1,...,n\}\rightarrow\{1,...,n\}$ we have the following: ...
1
vote
1answer
36 views

A characterization for subgroups.

Let $G$ be a group and $a_0,a_1,...,a_n\in G$ and $$A=\{a_0,a_1,...,a_n\}$$ and $$(\forall m\le n)(\forall i\le m)(a_{i}a_{m-i}\in A)$$ Is $A$ a subgroup of $G$? How if $G$ is abelian?
4
votes
1answer
59 views

Groups of order $p^aq^br^c$ containing elements of order $pq$, $qr$, and $pr$, but not $pqr$

Let $G$ be a solvable group of order $p^aq^br^c$ (for distinct prime $p,q,r$) containing elements of orders $pq$, $qr$, and $pr$, but no element of order $pqr$. Furthermore, assume that $G$ is ...
1
vote
0answers
52 views

Heineken and Mohamed Groups

I was trying to understand the costruction of the Heineken and Mohamed groups. (II) Every proper subgroup of $G$ is subnormal and nilpotent. Lemma 1. If $G$ is a non-nilpotent soluble group ...
7
votes
1answer
107 views

What is your favorite group? [closed]

I would like to know about your favorite group(s). Since groups do appear everywhere in mathematics and there are plenty of them, which ones have drawn your attention the most or surprised you? Please ...
3
votes
1answer
83 views

If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
-1
votes
4answers
100 views

Looking for a counter example - Normal subgroups and quotient group

I need to find a counter example for the following: Let $G$ be a group, and let $A,B\triangleleft G$ be two normal subgroups of $G$. if $G/A\cong B$ then $G/B\cong A$. Here are my thoughts so far: ...
11
votes
0answers
154 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
12
votes
6answers
382 views

If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? [duplicate]

I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers ...
3
votes
1answer
52 views

Let $G_1,G_2$ be groups with 2 subgroups respectively $H_1,H_2$ satisfying certain conditions, must $|G_1:H_1|=|G_2:H_2|$

Let $G_1,G_2$ be groups with two subgroups respectively $H_1,H_2$ such that there is a bijection $f:G_1\rightarrow G_2$ and $f|H_1$ is a bijection between $H_1,H_2$. Must $|G_1:H_1|=|G_2:H_2|$ ? ...
3
votes
4answers
139 views

Number of elements in a group and its subgroups (GS 2013)

Every countable group has only countably many distinct subgroups. The above statement is false. How to show it? One counterexample may be sufficient, but I am blind to find it out. I have ...
3
votes
2answers
114 views

Groups with $\wedge$-irreducible trivial subgroup

Suppose $G$ is a group satisfying the following condition: $$H \cap K = \{1\} \implies H = \{1\} \;\text{ or }\; K=\{1\}$$ for any two subgroups $H$, $K$, i.e. the trivial subgroup is ...
4
votes
2answers
176 views

Normal products of groups with maximal nilpotency class

Let $H$ be a nilpotent group of class $a$ and $K$ a nilpotent group of class $b$. If $H$ and $K$ are normal subgroups of a group $G$, then we know that $HK$ is a normal nilpotent subgroup of $G$ and ...
3
votes
1answer
321 views

Counter examples in group theory

Let $G$ be a finite group with normal subgroups $N_{1}$ and $N_{2}$. Find counter examples to the following statements 1) If $N_{1}\cong N_{2}$ then $G/N_{1}\cong G/N_{2}$ 2) If $G/N_{1}\cong ...
9
votes
1answer
269 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
1
vote
2answers
73 views

prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ when $n$ is odd

let $n$ be odd integer , prove that : $D_{4n} $ is isomorphic to $ D_{2n} \times Z_2 $ it's an example which the text proves ! but i can't understand any thing from the argument ! but i tried to ...
2
votes
1answer
64 views

For every monoid $M$ with zero is there a group $G$ such that $\mathrm{Grp}(G,G)\cong M$?

All monoids that I will consider here have identities. A monoid $M$ is said to have a zero iff $\exists z\in M \forall x\in M (zx=xz=z)$. Let $M$ be any monoid with a zero. Must there exist a group ...
2
votes
1answer
38 views

In a group, a finite left translation of $B$ covers $A$. Does any finite right translation of $B$ cover $A$?

Let $G$ be a group and $A,B\subseteq G$. Suppose there's some finite set $F\subseteq G$ such that: $$A\subseteq FB$$ Is there any finite set $F'\subseteq G$ such that $$A\subseteq BF'$$ ?
156
votes
21answers
14k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
1
vote
2answers
114 views

Examples of profinite simple groups

The standard example of an infinite simple group is $A_\infty$, the direct limit of the alternating groups under the obvious injections. Are there also examples of infinite simple groups arising as ...
1
vote
3answers
144 views

Formula for Product of Subgroups of $\mathbb Z$, Problem

What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$? Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, ...
27
votes
9answers
5k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
1
vote
1answer
88 views

Amenability of abelian and nonabelian groups.

Let $G$ be an abelian group. Is there any probability measure $\mu:\mathcal{P}(G)\to [0,\infty)$ such that for any $A\subseteq G$ and $x \in G$: $$\mu(A)=\mu(xA)$$ How if $G$ is not abelian? (do you ...
1
vote
0answers
36 views

Is there a nonvirtually abelian group whose commutator subgroup is finite? [duplicate]

Possible Duplicate: If the derived subgroup is finite, does the center have finite index? Let $G$ be a finitely generated group whose commutator subgroup is finite. Is $G$ necessarily ...
1
vote
0answers
46 views

Duality between $[G,G]$ and $Z(G)$? [duplicate]

Possible Duplicate: Center-commutator duality Let $G$ be a group. It seems that there is a certain duality between two of its normal subgroups, the commutator \begin{equation} ...
1
vote
1answer
148 views

On finite groups whose center is elementary abelian group

Let $G$ be a finite 2-group such that $Z(G)$ is elementary abelian 2-group ($\mid Z(G)\mid\geq 4$) and $Inn(G)$ is of order 4. Then prove that there exists an $\alpha\in Aut(G)$ such that ...
9
votes
1answer
351 views

Is there a counterexample to this weakened converse of Hall's theorem?

Suppose that a finite group $G$ contains a Hall $\{p,q\}$-subgroup for every pair of prime divisors $p,q$ of $|G|$. Does it follow that $G$ is solvable?
3
votes
1answer
139 views

On groups whose center has odd order

Let $G$ be a finite group such that $Z(G)$ is of odd order and $Inn(G)$ is of even order. Then prove $G\simeq Z(G)\times N$, such that $N$ is a subgroup of $G$ where $N\simeq Inn(G)$. Thank you!