The study of symmetry: groups, subgroups, homomorphisms, group actions.

learn more… | top users | synonyms (2)

4
votes
2answers
217 views

On the element orders of finite group

Let $G$ be a finite group. Suppose that $G$ has a normal Sylow $p-$subgroup $P$ such that $|P|=p^2$ where $p\neq 2$, but $P$ does not contain an element of order $p^2$ or equivalently $P$ is not ...
2
votes
1answer
215 views

Converting a (signed) permutation to a reduced word

I vaguely know that by looking at the inversions of a permutation, you can write down the reduced word expressing the permutation as a product of adjacent transpositions $s_i = (i,i+1)$. However, I ...
1
vote
4answers
212 views

How to write down explicit elements of a group from its presentation?

Let $$ G=\langle s_1, s_2, s_3 \mid s_1^2=s_2^2=s_3^2=1, (s_1s_2)^4=(s_2s_3)^3=(s_1s_3)^2=1 \rangle. $$ How to write down explicit elements of $G$? The following elements $$ 1, \ s_1, \ s_1s_2, \ ...
1
vote
3answers
130 views

How to write presentation of a group?

Let $G=(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$, where $(\mathbb{Z}/2\mathbb{Z})^n$ is the direct product of $n$ copies of $\mathbb{Z}/2\mathbb{Z}$, $S_n$ is the symmetric group of degree $n$, and ...
2
votes
2answers
776 views

What are differences between semidirect product and direct product?

Given two groups $A, B$, we can construct direct product $A \times B$ whose elements are of the form $(a, b), a \in A, b\in B$. If $A, B$ are subgroups of a group $G$ and $A \cap B =\{1\}$, then we ...
2
votes
3answers
133 views

Determine invertible and inverses in $(\mathbb Z_8, \ast)$

Let $\ast$ be defined in $\mathbb Z_8$ as follows: $$\begin{aligned} a \ast b = a +b+2ab\end{aligned}$$ Determine all the invertible elements in $(\mathbb Z_8, \ast)$ and determine, if possibile, ...
3
votes
2answers
124 views

What are the generators and relations for type $B_3$ Weyl group?

What are the explicit generators and relations for type $B_3$ Weyl group? Thank you very much. Edit: type $B_3$ Weyl group $G$ is $(\mathbb{Z}/2\mathbb{Z})^{3} \rtimes S_3$, so the order of $G$ is ...
5
votes
1answer
202 views

non-residually finite group

Let $G$ be the subgroup of $\text{Bij}(\mathbb{Z})$ generated by $\sigma : n \mapsto n+1$ and $\tau$ which switches $0$ and $1$. How can we prove that $G$ is not residually finite? Is it hopfian?
1
vote
1answer
131 views

Set of homomorphisms from discrete upper triangular group into continuous u.t. group

Let $G$ be the group $$ \begin{pmatrix} 1 & a_{12} & a_{13} & a_{14}\\ 0 & 1 & a_{23} & a_{24}\\ 0 & 0 & 1 & a_{34}\\ 0 & 0 & 0 & 1 \end{pmatrix} $$ ...
2
votes
0answers
247 views

Semidirect product group actions

$H$ and $K$ are groups, and $\Gamma$ is a set acted upon by H, while $\Delta$ is a set acted upon by $K$. Let $W := K \wr_\Gamma H$, the wreath product of $H$ and $K$. I have seen theorems stating ...
3
votes
2answers
229 views

This classic from euclid's elements, is it accepted everywhere?

I was reading linear vector spaces. When doing some exercise to prove some statements based on the properties defined for linear vector spaces, i suddenly noticed, outside the things defined, i'm ...
0
votes
2answers
108 views

Subgroups written as products

Suppose a finite group $G$ is the product of two of its proper subgroups $G=AB$. Assume also that $A\lhd G$ and that $A,B$ have relatively prime orders. Isn't it true that any subgroup $H$ of $G$ can ...
2
votes
1answer
127 views

Normalizer of regular action made linear

For a finite group $G$ the regular action $\rho$ of $G$ on itself (by right multiplication) has the property that the normalizer of $\rho(G)$ in the symmetric group $S_G$ is isomorphic to the ...
2
votes
3answers
139 views

number of automorphism on $\mathbb{Z}_9\times \mathbb{Z}_{16}$

we have to find number of automorphism on $\mathbb{Z}_9\times \mathbb{Z}_{16}$ I know a result which says $Aut(\mathbb{Z}_n)\cong U_n$ where $U_n$ is the multiplicative group i.e ...
1
vote
1answer
162 views

The number of cyclic subgroup

Let $p$ be prime divisor of order of finite group $G$, and the number of cyclic subgroup of order $p$ be $p+1$. If $P$ is a Sylow $p$-subgroup of $G$, then $P$ is normal in $G$ and $|P|=p^{2}$($P$ is ...
4
votes
2answers
256 views

A free group on the non-empty set $X$ is solvable iff $|X| =1$

Let $X$ be a non-empty set. Prove that $F_X$, the free group on $X$ is solvable if and only if $|X| = 1$. We can see that if $|X| = 1$, then $F_X$ is abelian, and hence solvable. However, the other ...
-3
votes
2answers
657 views

Non-isomorphic abelian groups of order $19^5$

I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?
2
votes
2answers
559 views

Group of order $60$

Let $G$ be a group of order $60$, pick out the true statements: a. $G$ is abelian b. $G$ has a subgroup of order $30$. c. $G$ has subgroups of order $2$, $3$, and $5$. d. $G$ ...
9
votes
2answers
352 views

Simple group of order $660$ is isomorphic to a subgroup of $A_{12}$

Prove that the simple group of order $660$ is isomorphic to a subgroup of the alternating group of degree $12$. I have managed to show that it must be isomorphic to a subgroup of $S_{12}$ ...
1
vote
0answers
48 views

Doubt in the proof of Conjugacy of positive systems under reflection group

I am stuck at a small thing in the proof of conjugacy of positive systems under a finite reflection group. I am using the notation and definitions used in the text by James E. Humphreys. I reproduce ...
1
vote
1answer
125 views

How to prove $\frac{G}{Z(G)}\cong \frac{\mathbb{Z}}{p\mathbb{Z}}\times \frac{\mathbb{Z}}{p\mathbb{Z}} $

Let $G$ be a group non-abelian group of order $p^3$, where $p$ is a prime number, prove that: $\fbox{1}$ $|Z(G)|=p$ $\fbox{2}$ $Z(G)=G'$ $\fbox{3}$ $\frac{G}{Z(G)}\cong ...
-3
votes
3answers
849 views

Any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic.

$\fbox{1}$ Prove that any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic. $\fbox{2}$ Prove that $\mathbb{Q}$ is not isomorphic to $\mathbb{Q}\times \mathbb{Q}$. Any hints would be ...
8
votes
1answer
191 views

Subgroups between $S_n$ and $S_{n+1}$

Lets look at $S_n$ as subgroup of $S_{n+1}$. How many subgroups $H$, $S_{n} \subseteq H \subseteq S_{n+1}$ there are ?
2
votes
2answers
108 views

Is there an unique “minimal enclosing group” for any two groups?

I'm not sure I'm using the correct terms, therefore let me define what I mean: Given a set of groups $G_i$, $i\in I$, I call an enclosing group of those groups any group $G$ so that for all $i\in I$ ...
8
votes
3answers
242 views

Group of groups

The product $\times$ of two groups is associative and commutative and there's a neutral element $\{1\}$. Let's say I create "virtual groups" which are inverses with respect to $\times$ (like getting ...
1
vote
1answer
667 views

Solving a polynomial modulo an integer

Say I have a polynomial $F$ of degree $n$ with coefficients in $Z_m$ and I wish to find $x$ such that $F(x)=0$ (mod $m$). For instance if $F(x)=x^{2}-a$ the solution would be the modulo $m$ squareroot ...
5
votes
2answers
199 views

Groups such that every finitely generated subgroup is isomorphic to the integers

What are examples of groups such that every finitely generated subgroup is isomorphic to $\mathbb{Z}$?
6
votes
5answers
691 views

Why is associativity required for groups?

Why is associativity required for groups? I'm doing a linear algebra paper and we're focusing on groups at the moment, specifically proving whether something is or is not a group. There are four ...
10
votes
0answers
154 views

Official name(s) for a certain type of p-group

I'm implementing a class of groups into Sage (sagemath.org), a computer algebra system, and I'm wondering if it has any official names. I found it in Gorenstein's "Finite Groups." It is there called ...
2
votes
1answer
255 views

Bounding Variance of a Convolution

Let $G$ be a group with finite subsets $A,B \subseteq G$. Let $k$ be an integer between $1$ and $|A|$ (or |A|/2 if it helps), and let $C$ be a random subset of $A$ of size $k$, chosen uniformly out ...
3
votes
2answers
255 views

Presentation of discrete upper triangular group

Let $G$ be the nilpotent Lie group consisting of matrices $$ \begin{pmatrix} 1 & a_{12} & \cdots & a_{1,n}\\ 0 & 1 & \ddots & \vdots\\ \vdots & \ddots & \ddots & ...
2
votes
3answers
251 views

Intertwiner in german?

What is the best way to translate the mathematical term ''intertwiner'' (between two representations of a group) into German?
0
votes
1answer
114 views

Can a subset of a group, which does not contain the identity element, be a group

Let $G$ be a group. Let $1 \in G$ be the identity element of $G$. Let $S \subset G$, with $1 \notin S$. Is it possible for $S$ to be a group, with some other element playing the role of the identity ...
4
votes
1answer
248 views

is this “adjoint” representation of $\mathfrak{gl}_2(\mathbb{F}_p)$ irreducible?

Let $\mathbb{F}_p$ be a finite field. There is an action (by conjugation) of $\text{GL}_2(\mathbb{F}_p)$ on the vector space of $2 \times 2$ matrices with coefficients in $\mathbb{F}_p$ that have ...
13
votes
0answers
409 views

Normalizers of automorphism groups

In abstract groups $\Gamma$ the normalizer $N_\Gamma(S)$ of a subset $S\subseteq\Gamma$ is the subgroup of all $x \in \Gamma$ that commute with $S$, i.e. $xS = Sx$, i.e. $x\ y\ x^{-1} \in S $ for all ...
0
votes
2answers
56 views

Group inclusion- Quotiens Inclusion

Given a group $G$ , and two subgroups $ G_1, G_2 $ such that $ G_1 \subseteq G_2 $ , is it true that $ G/G_2 \subseteq G/G_1 $ ? Thanks in advance !
2
votes
1answer
115 views

If $N\lhd H×K$ then $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially

I am thinking on this problem: If $N\lhd H×K$ then either $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially. I assume; $N$ is not abelian so, there is $(n,n')$ and $(m,m')$ in $N$ ...
0
votes
1answer
26 views

Is it possible to have a point $P_1$ not $\chi$-semistable but $P_2$ $\chi$-semistable with these two points in the same orbit?

Let $G$ be a group acting on an affine variety $X\subseteq \mathbb{A}_{\mathbb{C}}^n$. Suppose $P_1$ and $P_2$ are two points in $X$ such that $g\circ P_1=P_2$ for some $g\in G$. This means that ...
14
votes
2answers
311 views

Connectedness of centralizer exercise

I'm having trouble understanding connectedness from a group theoretic perspective. Let $G$ be the symplectic group of dimension 4 over a field $K$, $$G = \operatorname{Sp}_4(K) = \left\{ A \in ...
1
vote
1answer
171 views

Prove that G is cyclic if distinct subgroups have coprime orders

The order of the group $G$, meet the following conditions: $1<G<n$ where n is a natural number. For each 2 sub groups $H_1$, $H_2$ of $G$, if $H_1 \neq H_2$ then $\gcd(|H_1|,|H_2|)=1$. ...
1
vote
2answers
197 views

Maximal normal subgroup not containing an element

Do you know results about maximal normal subgroup among normal subgroups not containing a given element $x$ ? The problem can be reduce to the case of free groups. First, such a sugroup exists thanks ...
2
votes
1answer
226 views

Classify all groups of order $1805.$

Classify all groups of order $1805.$ It may help to note that $\left(\begin{array}{c} 0 &-1\\ 1 & 4 \end{array}\right)$ has order $5$ in $GL_2(\mathbb{F}_{19}).$ My idea: Observe that ...
0
votes
1answer
140 views

Semidirect product with a $p$ group

let $P$ be a finite $p$-group that acts on a finite group $G$ and assume that $P$ is maximal subgroup of $G \rtimes P$. Show that $G$ is an abelian $q$-group for some prime $q$. Hint: Show that $P$ ...
1
vote
1answer
112 views

numbers, matrices, vectors are groups? addition is possible between any combination?

Mathematical structures such as Numbers, Matrices, Vectors are all groups, rings, or similar? Can you provide more examples of more thease. Do operations such as addition occur (although in ...
0
votes
1answer
342 views

Automorphism of an Infinite cyclic group

$\newcommand{\Id}{\operatorname{Id}}$ $f$ is an automorphism of an infinite cyclic group $G$ then 1.$f^n\neq \Id_G$ 2.$f^2=\Id_G$ 3.$f=\Id_G$ if $f^n=\Id_G$ then every element of $G$ will have ...
1
vote
2answers
145 views

Finite non-abelian $p$-group cannot split over its center

Show that a finite non-abelian $p$-group cannot split over its center. I'd be happy for some clues.
0
votes
2answers
48 views

show that $\operatorname{cent}_{S_n}(h) \neq \langle h\rangle$

After counting the number of conjugates of $h=(1 2 \ldots n-2)$ I get $n(n-1)(n-3)!$ which when plugged in to the orbit stabilizer theorem gives the centralizer to have order $n-2$ but this would show ...
2
votes
2answers
214 views

Finite p-group with a cyclic frattini subgroup.

I have a question about the following theorem that I found in some research. Is it possible that $E$ is the identity? I just found this elaborated proof that might help.
1
vote
2answers
144 views

Group homomorphism to the multiplicative subgroup of a field

Let $G$ be a finite group and let $\varphi:G \rightarrow F^{\times}$ be an homomorphism where $F$ is a field. $H$ is a subgroup of $G$ that contains $Ker(\varphi)$. Prove that $H \lhd G$ and that ...
3
votes
1answer
397 views

Torsion-free quotient group of an abelian group

Let $G$ be an abelian group, and let $H\leq G$. Prove that if $G/H$ is torsion free, then $H$ contains the torsion group of $G$. Proof: Let $x\neq1$ be an element in the torsion group. Thus there ...