Questions about the branch of abstract algebra that deals with groups.

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66 views

### long root elements fixed by an automorphism in simple lie type group

Thanks for any help or comments.
Suppose $G=G(q)$ is a simple lie type group over field $F$ of characteristic $r$. So $G$ has some well known subgroup $X_a$ named Long root subgroup such that ...

**7**

votes

**1**answer

274 views

### For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group

For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then ...

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**0**answers

197 views

### Is always the ratio (number of commuting pairs of elements in a ring or a Lie algebra)/(the size of the ring or the Lie algebra) integer?

(1) Is there a finite nilpotent ring $R$ such that the ratio
$$c(R)=\frac{|\{(x,y)\in R\times R \; | \; xy=yx\}}{|R|}$$
is not integer?
Edit 1: The nilpotent condition is put later.
Edit/Answer: ...

**6**

votes

**1**answer

184 views

### Generators of pure braid groups of arbitrary Coxeter groups

Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma_i\in A$. Let $P$ be the "pure braid group", the kernel of the ...

**2**

votes

**1**answer

224 views

### Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the “other” Schur multipliers of a group?

The name for the the following 2 mathematical objects:
$$H_2(G,\mathbb{Z})$$ and
$$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times ...

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votes

**0**answers

119 views

### When can a locally compact group be approximated by discrete subgroups?

This question is about partitioning a (locally) compact group into cells by using discrete subgroups.
Let $G$ be a locally compact group. (I am really most interested in the case where $G$ is a ...

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votes

**1**answer

178 views

### homological 2 dimensional groups

In a Commentarii Mathematici Helvetici paper by Benno Eckman and Heinz Müller in 1980 (volume 50, pages 510-520) proved that poincaré Duality Groups of dimension 2 with positive first ...

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**0**answers

271 views

### What kind of group invariants exist? [closed]

Let $G$ be a finite group.Then it is known,
that:
1) The group determinant determines the group (up to isomorphism)
2) The 1, 2, 3 characters determine the group
3) The invariants $f_1,\cdots,f_m$ ...

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votes

**1**answer

365 views

### Perfect group of order 190080

I need to know some properties of the perfect group of order $190080$
which is the Schur cover of the Mathieu group ${\rm M}_{12}$, but
when using ...

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**0**answers

247 views

### Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...

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votes

**1**answer

452 views

### Why is $(A^\perp)^\perp = A$?

On page 52 of this paper, Iwasawa considered the bilinear symmetric non-degenerate pairing $\Phi_n \times \Phi_n \rightarrow \mathbb{Q}_p/\mathbb{Z}_p$ defined by
$$\langle \alpha, \beta \rangle_n := ...

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vote

**1**answer

195 views

### Generalization of $(HK:H)=(K:H\cap K)$

I asked this question two days ago om Math SE but didn't receive an answer: http://math.stackexchange.com/questions/1597321/generalization-of-hkh-kk-cap-h
Suppose we are given subgroups $H,K$ of a ...

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votes

**3**answers

320 views

### Frobenius complement/kernel of an infinite group

Happy Peaceful New Year !
In this question, I recalled that if $H$ is a proper subgroup of a finite group $G$, such that
$$({\bf A1})\qquad(g\not\in H)\Longrightarrow(g^{-1}Hg\cap H=(1)),$$
then
...

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votes

**2**answers

241 views

### Abelian extremely amenable group?

Is there a nontrivial commutative Hausdorff topological group that is extremely amenable?
Recall that a topological group is called extremely amenable if any continuous action on a compact ...

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**1**answer

111 views

### embedding of $O_4^-(q)$ in $U_4(q)$

For $q$ an odd prime power, one can construct $H:=O_4^-(q)<U_4(q)$ by treating the $H$-invariant bilinear form as Hermitean. On the other hand $U_4(q)$ is isomorphic to $O_6^-(q)$; I need to ...

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votes

**2**answers

475 views

### factorization of the regular representation of the symmetric group

Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation.
Question: Does there exist a representation $V$ (of dimension ...

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votes

**2**answers

155 views

### Frobenius Groups of Automorphisms

Recently, I am looking different papers on the topic
$$\mbox{Frobenius groups of automorphisms of a group.}$$
But I am familiar with Frobenius groups only; not with their (faithful) actions on groups. ...

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votes

**2**answers

324 views

### A question about generating set of groups and epimorphism

Do there exist non-isomorphic finitely generated groups, $G$ and $H$, along with epimorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$, such that every generating set of these groups is an ...

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**1**answer

160 views

### Does every generating set of the first homology group of a Cayley graph give rise to a presentation of its group?

Let $G$ be a group, and fix a symmetric generating set $S$. Let $X$ be the corresponding Cayley graph.
Let $R$ be a set of words in $S$, each corresponding to the identity of $G$, such that the set ...

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votes

**2**answers

237 views

### Behaviour of cohomology groups under extension of scalars

Let $\hat{R}\to R$ be a homomorphism of commutative unital rings and let
$\hat{M}$ be an $\hat{R}G$-module for a group $G$. Does the $R$-module isomorphism $$H^n(G,\hat{M}\otimes R)\cong ...

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**1**answer

95 views

### Characterization of some finite cyclic groups [closed]

Given a finite cyclic group $G$ with order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $p_i$s are distinct prime numbers $n_i>1$ for all $i$. Let $H$ be any abelian group. Assume that ...

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**1**answer

293 views

### Which finite groups can be characterized by their automorphism groups?

Given a finite group G, we denote by Aut(G) the group automorphisms of G . Which finite groups G can be characterized by the group Aut(G), i.e. Aut(H)≅Aut(G) implies H≅G?

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votes

**1**answer

191 views

### Characterizing cyclic group of order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, by Lattice isomorphisms

Given a finite cyclic group $G$ with order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $p_i$s are distinct prime numbers $n_i>1$ for all $i$. Let $H$ be any group.Assume that $L(G)≅L(H)$and ...

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votes

**1**answer

74 views

### Integer Triples and Reflection tiling $1,2,\ldots,n$

$\forall a,b\in\mathbb Z,\ \exists n\in \mathbb N$ such that the numbers $1,2,\ldots,n$ can be tiled using translates of $\{0,\ a,\ a+b\}$ and $\{0,\ -a,\ -(a+b)\}$ ?
In other words for every integer ...

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votes

**0**answers

167 views

### Manifolds as simultaneous coset spaces

Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of ...

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202 views

### Simultaneous coset spaces [closed]

Let $X$ and $Y$ be sets. Under what conditions is there a group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of a point in ...

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**1**answer

166 views

### Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?

Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and ...

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**1**answer

217 views

### 4th Order Floretions: Floret's Equation [closed]

Update: I've marked this question as answered. If you are thinking "What the heck are floretions?", go right to the answer provided by the Grinch. I definitely should have added clearer information ...

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votes

**2**answers

220 views

### Maximal size of minimal generating set

Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer ...

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**0**answers

108 views

### Which Coxeter groups can be realized as affine reflection groups?

Every affine reflection group has a Coxeter presentation (https://en.wikipedia.org/wiki/Reflection_group#Relation_with_Coxeter_groups). How do you tell which Coxeter presentations arise from affine ...

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71 views

### When can a 2-cocycle on a subgroup can be extended?

This question is based on a question when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective?
I am asking this as a new question as I already asked that user but got no ...

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59 views

### Almost invariance in compact quotients of locally compact groups

While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups
in the topological setting, I face the following problem.
Let $G$ be a locally compact amenable ...

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**0**answers

59 views

### Determining finite abelian groups among algebraic theories by counting

Let $T$ be a Lawvere theory (algebraic theory) which contains the theory $T_{\mathrm{grp}}$ of groups as a subtheory (so that $T$ has, in general, more equations that $T_\mathrm{grp}$).
Suppose ...

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**1**answer

317 views

### Can a large transitive permutation group need many generators?

let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have ...

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votes

**1**answer

102 views

### Perfect $Q[G]$-complex

Let $G$ be a finite group and let $M$ be a perfect $\mathbb{Q}[G]$-complex.
Suppose that $M\otimes_{\mathbb{Q}[G]}\mathbb{Q}$ is quasi-isomrphic to $0$ can we conclude that $M$ is quasi-isomorphic to ...

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**0**answers

202 views

### When is the equalizer of group homomorphisms a normal subgroup?

Let $\phi,\psi:G\to H$ be a pair of group homomorphisms. Their equalizer is the subgroup $C\le G$ given by
$$
C=\{ g\in G \mid \phi(g)=\psi(g)\}
$$
My question is:
Under which conditions is the ...

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votes

**1**answer

209 views

### When the commutativity of the product of group subsets implies the commutativity of the group?

Let $G$ be a finite group. Which are the values of $k$ for which if every two $k$-subsets of $G$ commute then $G$ is commutative? Clearly, this holds for $k=1$ and $k=2$.

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230 views

### Non-congruence normal subgroups of $SL_2(\mathbb{Z}[1/2])$

Let $G=SL_2(\mathbb{Z}[1/2])$, i.e., the modular group (if you wish) over the ring $\mathbb{Z}[1/2]$ consisting of rationals whose denominators are powers of $2$. Unlike $SL_2(\mathbb{R})$, $G$ is ...

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**0**answers

115 views

### Acyclicity of covering space

Suppose we have some 2-dimensional non-aspherical finite CW-complex $K$ with $\pi_1(K)=G$. Is there any sufficient condition on $H\leq G$ (and maybe on the group $G$ itself) which allows to conclude ...

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**4**answers

576 views

### When does $\langle grg^{-1}|g\in G\rangle = \langle gsg^{-1}|g\in G\rangle$ imply $\langle r\rangle=\langle xsx^{-1}\rangle$?

We call a finite group $G$ normal if for all $s,r\in G$, if $\langle gsg^{-1}:g\in G\rangle =\langle grg^{-1}:g\in G\rangle $ then there exists $x\in G$ such that $\langle r\rangle =\langle ...

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votes

**3**answers

955 views

### Can Calabi-Yau manifolds have nonabelian discrete symmetry groups?

A particle physicist asked me the above question. Let me try to make it more precise. Suppose $M$ is a 3-dimensional Calabi-Yau manifold: that is, a compact Kähler manifold of complex dimension ...

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**1**answer

308 views

### What do we know about these subgroups of $S_n$?

For each positive integer $n$, write $S_n$ for the symmetric on $n$-letters. Suppose that $m | n$ is a proper divisor of $n$, and write $n = km$. Consider the element
$$\displaystyle u(m,n) = ...

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82 views

### Group which is not MF or AF

Does someone know example of group (countable, discrete) which can not be embedded (monomorphism) into
$$ U(\prod M_n/\oplus M_n)$$
unitary group of universal MF-algebra? Or example of group which can ...

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votes

**1**answer

97 views

### Notation for a Homomorphism [closed]

Is there a (common) notation which denotes a function, $f$, to be a homomorphism?
I have found myself writing, "let $f: X \rightarrow Y$ be a homomorphism" several times. This is fine, but I would ...

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67 views

### Distal actions on coset spaces

Let $H$ be a group acting by homeomorphisms on a Hausdorff space $X$. Say the action is distal if for all $(x,y) \in X \times X$, if the set $\{(hx,hy) \mid h \in H\}$ accumulates at a diagonal point ...

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73 views

### Correlation of Class Functions

Let $G$ be a finite group, and let $f_1,f_2$ be two real-valued class functions of $G$. Assume that multiplying elements of $G$ takes $O(1)$-time.
Let $s:G\to \mathbb{R}$ be defined by ...

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votes

**1**answer

129 views

### How to construct a proper action of a group of finite virtual cohomological dimension?

Let $\Gamma$ be the semidirect product of $\mathbb{Z}$ and $\mathbb{Z}/4$,
where the action of $\mathbb{Z}/4$ on $\mathbb{Z}$ is defined by $\bar{k} \cdot x = (-1)^k x$. Clearly $\Gamma$ has virtual ...

**1**

vote

**1**answer

121 views

### Are rotational isometries groups generated by some kind of rotations?

If we consider the index $2$ subgroup of a Weyl group consisting of the isometries with determinant $1$ (the 'special' Weyl group), is it known that it is generated by rotations around some fixed ...

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votes

**2**answers

432 views

### What categorical property of monoidal categories picks out the ones with duals?

Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...

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**0**answers

182 views

### Grothendieck - A group as a sheaf over simplicial complexes

In this blog post, Terence Tao gives the following definition of a group.
Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...