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

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0
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0answers
20 views

Direct product between joins of subgroups

Suppose that $G$ is a finite group with $A, B, H, K \leq G$. Suppose that $H\times A \leq G$ and $K\times B \leq G$. I want to show that $\langle H,K \rangle \times \langle A, B \rangle = \langle H \...
1
vote
0answers
21 views

Extending homomorphisms into Hahn groups

Let $\Omega$ be a linearly (i.e. fully) ordered set, and let $\Lambda_{\Omega}$ be the ordered abelian group consisting of those $(\lambda_\omega)_{\omega\in\Omega}\in\mathbb{R}^{\Omega}$ with well-...
8
votes
1answer
187 views

Groups whose word problem can be solved in constant time

Given a finitely generated group $G$, define an encoding of $G$ to be a one-to-one function $\Phi:G\to \bigcup_n \{0,1\}^n$ that sends each group element to a unique finite word. For $a,b\in G$, ...
3
votes
1answer
200 views
+50

Intuition behind the definition of the Siegel-Eichler transformation

Recently I am reading Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in. Let $X$ be an ...
3
votes
1answer
128 views

Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups

Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence". Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be ...
1
vote
0answers
74 views

Induced structure of topological group [on hold]

If we consider a closed Jordan curve $\mathcal{C}$, I know that it's homeomorphic to the circle $S^1$. Now I take an homeomorphism $\phi:S^1\longrightarrow\mathcal{C}$ and this homeomorphism induces a ...
1
vote
1answer
59 views

Non-trivial homomorphisms to finite groups on fixed generating set

A group $G$ is residually finite if for each element $g\in G$ there exists a (surjective) homomorphism $f_g: G \rightarrow H_g$ such that $H_g$ is finite and $f_g(g)\ne 1$. Consider the weaker ...
4
votes
1answer
190 views

Mapping a group to a finite group s.t. the image of each generator is nontrivial

Recall that a group $G$ is called residually finite if for any nontrivial element $g\in G$ there exists a finite group $H$ and a homomorphism $f$ from $G$ to $H$ such that $f(g)\neq1$. My question is ...
2
votes
1answer
130 views

Size of automorphism group of random regular graph

If I pick a random regular graph on $n$-vertices and degree $d$ from uniform distribution what is the probability that its automorphism group is of size at least $m$? -- I want to know what is the ...
17
votes
0answers
283 views

Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with ...
-4
votes
0answers
71 views

How do I go about solving the following problem? [closed]

Given $(m_1,m_2, ...,m_r)\in Z^r_{\geq 0}$, and $a_1, a_2, · · · , a_r \in \mathbb{N}$ such that: $\sum_{i=1}^r a_im_i=qL$ where $L$ denotes the least common multiple of $a_1, a_2, · · · , a_r$ and $...
7
votes
3answers
197 views

When can the Cayley graph of the symmetries of an object have those symmetries?

Let $P$ be an object in $\mathbb{R}^n$ with symmetry group $G$. Let $C$ be the a Cayley graph of $G$. When can $C$ be embedded in $\mathbb{R}^m$ so that the embedded graph has the same symmetry ...
2
votes
0answers
87 views

example of fuchsian groups acting on 2-sphere by G. Martin

Currently I am reading a paper "Infinite group actions on spheres" by Gaven Martin. I am a first year graduate students and I got lots of questions, so one of them is about the following example: (...
-4
votes
0answers
97 views

Finite groups are isomorphic [closed]

For two finite groups $G_1, G_2$ if for every integer $n\geq 0$, $|G_1^n| = |G_2^n|$, then is it true that $G_1\cong G_2$? By $G^k$ we mean set $\{g^k|g\in G\}$.
4
votes
1answer
95 views

A quotient group of a self-normalizing spherical subgroup

Let $G$ be simply connected, simple algebraic group over $\mathbb{C}$. Let $H\subset G$ be a self-normalizing spherical subgroup of $G$, not necessarily connected or reductive. Here "self-normalizing" ...
9
votes
1answer
228 views

The set of subgroups of $F_2$

This question came up in our algebraic topology class and our Professor didn't know the answer. I also couldn't find an answer so far. What is the cardinality of the set of subgroups of $F_2$? ...
17
votes
2answers
418 views

When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$?

A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, ...
2
votes
0answers
72 views

When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$. By Example of a Schur-nontrivial group with no abelian subgroup ...
1
vote
2answers
95 views

Finite generation of stabilizers in a $G$-set [closed]

Suppose that $G$ is a finitely generated group, $X$ is a $G$-set, and $x \in X$ is a point. Are there any sorts of conditions on $X$ and $G$ that would let me conclude that $\operatorname{Stab}(x)$ ...
6
votes
2answers
244 views

Simple groups and irreducible characters of degree 3

I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here. The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}...
3
votes
1answer
124 views

p-groups with maximal class subgroup

Suppose $G$ is a finite non-abelian p-group of nilpotent class $c$. Is there a subgroup $H$ of nilpotent class $c$ and size $p^{c+1}$? If this is not true, is it possible to add some additional ...
2
votes
2answers
117 views

Malnormal subgroups in solvable groups

I have been reading through a book of Robinson where it is mentioned (informally!) that solvable groups have "many" subnormal subgroups (subgroups $H<G$ with $H=H_0 \lhd H_1 \lhd \ldots \lhd H_n = ...
8
votes
1answer
135 views

Finite-index iff positive density?

Let $G$ be a finitely generated group and $S$ a symmetric generating set. Define density (lower density, say) with respect to the sequence of balls $S^n$. Is it true that a subgroup of $G$ has ...
3
votes
0answers
56 views

Group of Lie type as expanders: explicit estimates

In a paper Finite simple groups as expanders by M. Kassabov, A. Lubotzky and N. Nikolov there is a theorem, stating that there exists $\varepsilon>0$ and $k\in\mathbb{N}$, such that for every non-...
1
vote
1answer
130 views

p-groups with special property on its centralizers

Thanks for any help or comment. Suppose $G$ is a finite non-abelian p-group. Suppose $G$ has a proper non-abelian subgroup $M$ such that for every non-central element $x\in M$, $C_G(x)\subseteq M$. ...
10
votes
3answers
435 views

Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products

A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the reflector. If the category $C$ is ...
2
votes
0answers
109 views

Analogues of Goursat's Lemma for Infinite Products of Topological Groups

Recall that Goursat's Lemma has the following useful consequence. Let $G_1, G_2$ be finite groups with no common simple non-abelian quotients, and suppose $\gcd(|G_1^{\operatorname{ab}}|, |G_2^{\...
1
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0answers
154 views

The number of fixed points of an automorphism of $\mathbb{Z}_m\times\mathbb{Z}_n$

Let $m$ and $n$ be two positive integers such that the groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have no common direct factor. Then an automorphism $f$ of $\mathbb{Z}_m\times\mathbb{Z}_n$ is of type $$\...
5
votes
1answer
103 views

Bounding the union of conjugates of a maximal subgroup of the Symplectic group over a finite field

Let $g \geq 1$ be a positive integer, and let $p$ be a prime. Consider the symplectic group $G := \operatorname{Sp}_{2g}(\mathbb{F}_p)$ of symplectic matrices with entries in $\mathbb{F}_p$. Let $M \...
3
votes
3answers
566 views

A question on linear groups

Asking this question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now ...
1
vote
0answers
81 views

Geography of Kähler manifolds

What is the geography of Kähler manifolds with negative sectional curvature? More precisely, can any hyperbolic group be realized as the fundamental group of a Kähler manifold with negative sectional ...
4
votes
1answer
156 views

Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$

It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise ...
1
vote
1answer
118 views

Subsets of the boundary of a surface group

Consider the surface group $\Gamma=\langle a,b,c,d\mid [a,b][c,d]=1\rangle$: it is a Gromov hyperbolic group; its Gromov boundary $\partial\Gamma$ is homeomorphic to $S^1$ (the unit circle). I would ...
4
votes
0answers
59 views

Do the normal forms for braid groups really conceal information about better than randomly applying the braid relations?

In braid based cryptography, one typically wants to conceal the way a certain braid $b$ has been obtained. One therefore puts $b$ into some normal form. Since every braid has a unique normal form, the ...
0
votes
0answers
132 views

Ring structure on cohomology of groups

Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter ...
2
votes
1answer
145 views

Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes: $$ \forall f,g,h\in G:hg(f)=h(g(f)) $$ Now suppose there is additional axiom, or constraint if you prefer, ...
4
votes
2answers
212 views

Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups" https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...
6
votes
0answers
49 views

Coarse embeddability into Hilbert space of residually finite groups

By definition a finitely generated group G is coarsely embedded into Hilbert space if there is a function $F: G\to \ell_2$, such that $\|F(g_n)-F(h_n)\|\to\infty$ iff $d(g_n,h_n)\to\infty$, where $d$ ...
6
votes
1answer
153 views

Does every cocompact lattice admit a homomorphism (with infinite image) into a compact Lie group?

Let $\Gamma$ be a cocompact arithmetic lattice in a semisimple algebraic group. Does it admit a homomorphism $\Gamma \to K$ with infinite image into a compact real Lie group $K$?
1
vote
3answers
173 views

Finite subgroups (not finite index, just finite) of the modular group

The modular group is commonly described as the group of linear fractional transforms $z \mapsto \displaystyle \frac{az+b}{cz+d}$ with $a,b,c,d$ integers and $ad-bc = 1$. Of course, a great deal is ...
2
votes
2answers
321 views

Which is better for creating tables of group theory info, GAP or MAGMA?

Specifically, I want to compute the set of values of $|G:\text{ker}(\chi)|/\chi(1)$ for all the characters of a p-group, for a lot of p-groups. I don't know how to use either program, so before I ...
8
votes
1answer
240 views

Do you know this Burnside ring module?

Let $G$ be a finite group and $\Omega(G)$ its Burnside ring. There is a certain $\Omega(G)$-module, let's call it $M(G)$, that appears in something that I am thinking about. As an abelian group $M(G)$ ...
6
votes
0answers
217 views

On describing a sort of “well-behaved” subgroups of a free abelian group

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
2
votes
1answer
122 views

Coboundary for the Cohomology of free groups

Let $G$ be a group. Let $\Bbbk$ be a field of char. $0$. We denote with $C^{n}(G, \Bbbk)$ the set of maps $f\: : \: G^{n}\to \Bbbk$ and with $\partial_{G}\: : \: C^{n-1}(G, \Bbbk)\to C^{n}(G, \Bbbk)$ ...
12
votes
1answer
136 views

The finiteness criterium $F$ under quasi-isometry

A group $G$ is defined to have $F$ if there exists a finite $K(G,1)$. This property is clearly not invariant under quasi-isometry as one can see from the trivial group and $\mathbb{Z}_2$. My question:...
4
votes
2answers
236 views

Is $(G\rtimes H,H)$ a Gelfand pair iff $G$ is abelian?

Let $G$ be a finite group and $K \subset G$ a subgroup. Then $(G,K)$ is a Gelfand pair if the double coset Hecke algebra $\mathbb{C}(K \backslash G / K)$ is commutative. Let $H$ be a subgroup of $...
3
votes
2answers
143 views

Do the irreducible modules of this finite group preserve a tensor product structure?

I am interested in a particular group $G$, where $$ (A_4\times C_\ell) \lhd G \lhd S_4 \times D_\ell$$ Here, $C_\ell$ is cyclic, $D_\ell$ is dihedral of order $2\ell$, and the two inclusions both have ...
1
vote
0answers
51 views

Computing subgraph orbits

I have group $G$ acting on a 4-regular 120 node graph $\Gamma$. I want to compute equivalence classes of connected subgraphs of $\Gamma$, where by equivalent I mean in the same orbit. More ...
18
votes
1answer
623 views

Have finite doubly transitive groups been classified?

I am trying to determine whether the literature contains a complete proof of the classification of finite 2-transitive groups. This is a fundamental result with important applications in many areas ...
4
votes
1answer
266 views

Do irreducible characters form a closed set?

A character on a discrete group $\Gamma$ is a conjugation-invariant function $\tau$ which is of positive type, and is normalized so that $\tau(e) = 1$, where $e$ is the identity element of $\Gamma$. A ...