A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Can “anti-groups” and “anti-manifolds” be constructed? (and other “anti-objects”)

Is it possible to create an "antigroup"? What I mean by this is, given some group G, and some "antigroup" H, then the "free product" of G and H, G*H will equal the "group" (vacuously a group) of no ...
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Subgroups of $\operatorname{SL}(2,R/I)$ for $R$ local as $I$ varies

The rough question is: How do the finite subgroups of $\operatorname{SL}(2,R/I)$ vary as $I$ varies, where $R$ is a nice local ring? $$\operatorname{SL}(2,A) = \left.\left\{ \left(\begin{...
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How to get a Presentation of a Group

$\newcommand{\R}{\mathbf R}$ Let $G$ be the group of homeomorphisms of $\R^2$ generated by $g$ and $h$, where $g(x, y)=(x+1, y)$ and $h(x, y)=(-x, y+1)$. To show that $G\cong \langle a, b|\ b^{-1}...
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Kazhdan's property (T) vs residual finiteness

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $(F)$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with ...
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Minimal Normal Subgroups of an elementary abelian p-group

Can you explain/prove, why the number of minimal normal subgroups of an elementary abelian $p$-group of order $p^n$ (for instance of $\mathbb{Z_p}^n$), is exactly $(p^n-1)/(p-1)$? I know that it seems ...
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35 views

Show that $\phi(N) \leq H$

Let $\phi: G \to H$ be a group homomorphism and $N \leq G$ (with $G$, $N$ and $H$ groups). Show that $\phi(N) \leq H$ So this is what I did: Obviously $\phi(N)$ is a subset of H because $N$ is a ...
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37 views

The Cardinality of a subset of field of 8 elements

Let $F$ be a field of $8$ elements. Let $A$ be a subset of $F$ and $A = \{x \in F \mid x^7=1$ and $x^k \neq 1$ for $k<7\}$. Then find the number of elements in $A$. Argument: Since $F$ is a ...
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39 views

if HK is a subgroup for all K, does it imply that H is normal?

we know that in a group G, if H,K be subgroups such that H is normal, then the product HK is also a subgroup. does the converse hold? i.e. if H is a subgroup of a group G such that for any subgroup K ...
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27 views

Cayley table of a group of order n with 2n entries deleted

Let $G$ be a group of order $n$, on elements $x_1, ..., x_n$. Let's draw the Cayley table of the group, that is, table $A$ with $a_{i,j}$ = [number of $x_i \circ x_j$ in the list $x_1, ..., x_n$] ...
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Group such that Aut(G) = H and Aut(H) = G

So this question just popped into my mind: does there exist a group $G$ such that $Aut(G) = H$ and $Aut(H) = G$ and $H\not=G$. My intuition says no, but im not sure how to go about proving this. Can ...
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66 views

For which numbers $n$ does every group with order $n$ have a non-trivial center?

Which numbers $n$ have the property that every group with order $n$ has a non-trivial center ? $n$ has this property if it is an abelian number (every group with order $n$ is abelian). If $n$ is a ...
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36 views

Finite Fields and Discrete Log Problem

The finite field $\mathbb F_{p^p}$, for a prime $p$, can be described as $\mathbb F_p[x]/(x^pāˆ’xāˆ’1)$. Let $G=\mathbb F_{p^p}^\times/\mathbb F_{p}^\times$. Estimate the cost of obtaining the discrete ...
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40 views

If $|G_1|=|G_2|<\infty$ and $|G_1'|<|G_2'|$, then $|Z(G_1)|\geq |Z(G_2)|$? where $G'$ is the commutator subgroup of $G$.

We know that $G'$ characterization how ``abelian'' of a group because we have a theorem: if $G'=\{e\}$, then $G$ is abelian. I have a conjecture. If there are two finite groups $G_1$ and $G_2$, $|...
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53 views

Prove that $\phi$ is class preserving automorphism

Let $G$ be a finite group and $\phi:G\to G$ be an automorphism of $G$ such that $\phi|_P=conj(g)|_P$ (restriction of some inner automorphism of $G$) where $P$ is any sylow $p$-subgroup of $G$, then is ...
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47 views

Abelian subgroup of standard wreath product

Let $A$ and $B$ be non-trivial groups. We construct their (restricted) wreath product as follows. Denote by $A^{(B)}$ the set of all function from $B$ to $A$ with finite support, and equip it with ...
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Prove that there exist a sylow subgroup of $G$ which is fixed by $\alpha$.

Let $G$ be a finite group and $\alpha\in Aut(G)$ such that $\alpha$ restricted to any sylow subgroup of $G$ equals the restriction of some inner automorphism of $G$ and $(o(\alpha),o(G))=1$, then is ...
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64 views

Crossed homomorphism

Let $G$ be a group and denote $x^y$ by $yxy^{-1}$. A map $f\colon G\rightarrow G$ is called crossed homomorphism if $f(ab)=f(a)^bf(b)$. Question: In some matrix groups, can one give a nice example ...
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Minimal number of generatos in a pushout/Generalization of Grushko's Theorem

Let $A_1 \hookleftarrow \mathbb Z \hookrightarrow A_2$ be a diagram of finitely-generated, torsion free groups, where each arrow indicates an injective homomorphism. Let $A$ be the pushout of this ...
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A “new” formula relating the quotients of the upper central series, method of proof and background information

For a finite group $G$ the upper central series is defined inductively by $$ Z_1(G) := Z(G), \qquad Z_{i+1}(G) / Z_i(G) = Z( G / Z_i(G) ). $$ Now I am interested in generalising this formula, i.e. ...
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38 views

Sharply $2$-transitive subgroup of a group

My question is about the problem 2.8.14 of the book "Permutation Groups" by Dixon and Mortimer. Let $F=F_q$ be the finite field with $q$ elements and $d\ge 1$. For each linear transformation $a \in ...
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101 views

What is the smallest squarefree number $n$ with $gnu(n)=79$?

I am searching the smallest squarefree number $n$ with $gnu(n)=79$. $n$ must have more then $3$ prime factors because $p+2\ne 79$ and $p+4\ne 79$ for every prime $p$. The maximum possible number of $...
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Normal subgroup $H$ of $G$ with same orbits of action on $X$.

I have a somewhat broad question related to group actions and their restriction to a normal subgroup. If we have a group action $\sigma : G \times X \rightarrow X$ with orbits $G_x$, and a normal ...
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46 views

A name for elements of a group generating the same cyclic subgroup

Elements with similar properties usually deserve a name in many contexts, say primitive elements in finite fields, integers modulo a number $n$, generators of a free groups etc. Does there exist a ...
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63 views

$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set $\...
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42 views

Proving an eliptic curve is cyclic, and determining it's order

I need a solution with an explanation for the following. Thanks! Let $E/F_q$ be an elliptic curve and let $P āˆˆ E(F_q)$ be a point a. if $n=ord(P)>1/2(q^{0.5}+1)^2$ prove that $E(F_q)$ is cyclic ...
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Is any pair of finite 2-generated perfect groups the quotient of a third finite 2-gen perfect group?

Let $H_1,H_2$ be finite 2-generated perfect groups. Does there exist a finite 2-generated perfect group $G$ which has $H_1,H_2$ as quotients? Of course we may assume $H_1\not\cong H_2$. If $H_1,H_2$ ...
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Union of conjugates of a subgroup of a finitely generated group.

Is there a finitely generated group $G$, with a proper subgroup $H$, such that $$G=\bigcup_{g\in G}gHg^{-1}$$ I know that this is not the case for a finite $G$: Union of the conjugates of a proper ...
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About centralizers of involutions in finite simple groups

I read that for the classification of the finite simple groups the centralizers of involutions plays a crucial role. This has to to with the Brauer-Fowler-Theorem, which in one formulation could be ...
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Can we extend the group action from subgroups?

Let $G$ be a group and $H,K$ be subgroups of $G$ such that $G=<H,K>$. Suppose that $H,K$ acts on the set $S$. Is there any condition that which guarantees that action of $H,K$ is extended to ...
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63 views

Why do we represent groups in the category of vector spaces?

For any category $\mathcal{C}$ and any object $X$ in $\mathcal{C}$, $\operatorname{Aut}_{\mathcal{C}}(X)$ is a group. Thus, given any group $G$, it makes sense to talk about representing $G$ in $\...
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$\otimes$-Categorical Generalization of Lagrange

I am reading M. Brandenburg's paper and came across the following result which is a generalization of Lagrange's theory in group theory: Let $\mathcal C$ be a $\otimes$-category and $A\to B$ a ...
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Proving whether the following are groups or not.

In each case, I am asked to decide whether the indicated pair is a group or not. If so, prove it; if not, show which group axiom fails. (a) $(\dfrac{1}{2}\mathbb{Z}, +)$ where $\dfrac{1}{2} \mathbb{Z}...
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Proving $\mathrm{Aut}(S_{n}) = S_{n}$ for $n > 6$

This is a question for a school assignment. We are being asked to prove that $\mathrm{Aut}(S_{n}) = S_{n}$ for $n > 6$. These are the steps we are supposed to follow in our proof. Prove that an ...
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Geometric Coset

I am familiar with cosets, but im not sure what to do here (never had to give a "geometric" description of a coset): Consider $\mathbb{R}$ and the subgroup $\mathbb{Z}$. Describe a coset $t+\mathbb{...
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Elementary consequences of commuting limits and colimits over groups

In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is mentioned: Theorem 1. $H$-limits commute ...
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Symmetry group of product polytope

The symmetry group of the interval $[-1,1]$ is $\mathbb Z_2$, since it consists only of the identity and the reflection at the origin. Consider now the square $[-1,1]^2$. Obviously, its symmetry ...
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Question on special unitary group $SU(3)$ over local $p$-adic field

I am reading Casselman and Shalika's paper, the unramified principla series of p-adic groups II. I have a problem on the special unitary group $SU(3)$ over local p-adic field. Let $F$ be a local ...
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Is the free product $\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}$ a linear group?

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I want to know if $H$ is a ($\mathbb{Z}$)linear group that is to say is there an injective homomorphism $f: H\to GL_m(\mathbb{Z})$ ...
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Infinite group with unique maximal subgroup

If $G$ is a group, then by a maximal subgroup, we mean a proper subgroup, which is maximal w.r.t. subset(subgroup) relation. It is well known that a finite group with unique maximal subgroup must be ...
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Why $ K \cap H $ is a maximal subgroup of $H $?

Suppose $ G $ is a finite group and $ H \unlhd G $. Suppose that $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup $ M $ of $ G $. Let $ N $ be a minimal normal ...
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Infinitely iterated square roots in groups

Let $G$ be a group. What are possible conditions on $G$ to ensure that there is no sequence $\{g_i\}_{i\in\mathbb Z}\subset G\backslash\{1\}$ such that $g_{i+1}=g_i^2$ for all $i\in\mathbb Z$? Does ...
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79 views

$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h \...
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Proving a version of the Kronecker's Theorem

I am trying to prove the following version of the Kronecker's Theorem: Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then $\...
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Haar measure on a profinite group is the inverse limit of the counting measure on its quotients?

I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to: Let $N_i$ be a basis normal subgroup neighborhoods of the identity in a ...
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Is the monster group a characteristic quotient of $F_2$?

Let $F_2$ be the free group on two generators, and $M$ the monster group. It's known that every finite simple group is 2-generated, so let $F_2\rightarrow M$ be a surjection with kernel $N$. Let $...
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Finite conjugacy classes in a certain group with three generators

Def. We say that group G has non-trivial finite conjugacy class if there is a conjugacy class $C=\lbrace g_i \rbrace$ such that $g_i \neq 1$ of G with $|C|<\infty$. Let G be group $$<a,b,c \...
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Number maximum of commutators required to generate an element of the derived subgroup

Let $G$ be a group for which the center $Z(G)$ is of index $n$. How to prove that an element of the derived subgroup $G^\prime$ is the product of at most $n^3$ commutators?
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107 views

Free groups: normal supplements of the commutator subgroup

Let $F$ be a free group and let $V$ be another verbal subgroup of $F$ such that $$ F = [F,F] V. $$ Is it true that $V=F?$ More generally, if $N$ is a normal (or even characteristic) subgroup of $F$ ...
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63 views

Basic application of Weyl-Character-Formula

(I did not find a solution of my problem in any forum so far. Sorry if it exists...) I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only ...
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Is a finite group which is generated by two fully invariant abelian subgroups always abelian?

Let $G$ be a finite group satisfying there exist two fully invariant subgroups $H,K$ of $G$ such that $H$ and $K$ are abelian and generate the whole group $G$. Then can we conclude that $G$ is abelian?...