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.

learn more… | top users | synonyms (2)

2
votes
4answers
54 views

Find two finite abelian groups of same order $G$ e $H$ such that $G \ncong H.$

Find two finite abelian groups of same order $G$ e $H$ such that $G \ncong H.$ I found groups $\mathbb{Z}_4$ e $V$ (Klein's group) that satisfy it, but would like more examples. I'm trying to use the ...
0
votes
1answer
8 views

Compute variation left action subgroup

I consider a Lie group $G$, with a group element $g$ parametrised in some manner with parameter $\theta_i$, $i=1,\cdots, \dim G$. Suppose that $K\subset G$. I want to compute the variation of an group ...
3
votes
1answer
32 views

Geometric Representation of Quasidihedral Groups

I am going back through Dummit/Foote studying for a prelim and came across the 'quasidihedral' or 'semi-dihedral', group of order $2^n$, with presentation $\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs ...
1
vote
1answer
29 views

Why is $\operatorname{Hom}_{\mathrm{Groups}}(G,A)$ isomorphic to $\operatorname{Hom}_{\mathrm{Ab}}(G/[G,G],A)$?

This question is inspired by an exercise from the Weibel's book on Homological Algebra (beginning of chapter 6 on Group Cohomology). Let $G$ be a group and $A$ be a $G$-module. My question simply is: ...
3
votes
1answer
44 views

Looking for group of polynomials with only real roots

Assume $P_\mathbb R$ is the set of all polynomials which have only real coefficients and only real roots. Define $0$ as a polynomial with infinitely many real roots and all other constant polynomials ...
4
votes
1answer
48 views

Induction of an irreducible group representation

I'm having some trouble finding the answer to the following question. Any ideas on how to get started? Let $H$ be a subgroup of a group $G$ and let $U_{1}$, ...,$U_{k}$ be the irreducible ...
1
vote
1answer
48 views

$SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong \mathbb A_4$

I am trying to prove that the quotient group $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3))$ is isomorphic to $\mathbb A_4$. I could show that $SL_2(\mathbb Z_3)$ has $24$ elements (one can see this ...
4
votes
0answers
61 views

Help with translating from French to English

In a paper by G. Boccara, "Cycles comme produit de deux permutations de classes données," I have come across something that seems weird. On page 130, notation 1.7, it says: Si $l$ est un entier ...
2
votes
1answer
22 views

2-Frobenius groups of order $2^{10}.3^5.5.11$

A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with ...
0
votes
0answers
9 views

Partial Group with elements only comparable to 0?

The use case is for managing a digital ledger where the monetary amounts are kept private, but transactions are still verifiable through addition. To do this, I need a mathematical construct which ...
6
votes
0answers
89 views

(Non-Hopfian) groups that only have quotients that are themselves or the trivial group.

A group is non-Hopfian provided it is isomorphic to a proper quotient. The classic, finitely presented, example of such a group is the Baumslag-Solitar group $$BS(2,3)= \langle x,t \mid t^{-1}x^2 t ...
3
votes
1answer
75 views

If $C_G(H)=N_G(H)$ for all abelian subgroups, prove that $G$ is abelian

Let $G$ be a finite group such that for all abelian subgroups $H$ of $G$, $$C_G(H)=N_G(H).$$ Prove that $G$ is abelian. ($C_G(H)$ is the centralizer, $N_G(H)$ is the normalizer of $H$ in $G$) my ...
2
votes
0answers
59 views

Does group $G$ of order 42 have a normal cyclic subgroup of order 21?

Show that a group $G$ of order 42 has a normal cyclic subgroup of order 21. What I did so far is using Sylow's theorem to show that $G$ has a unique 7-sylow subgroup $S(7)$ (which is normal) and {1 ...
1
vote
1answer
36 views

Count number of colorings of tetrahedron, where colorings are indistinguishible if one can be reached from another by rotation

I'm fairly new to group theory, and here's one problem I'm trying to solve: We're coloring nodes of tetrahedron in 3 distinct colors, and its edges in 2 distinct colors. We're treating two colorings ...
1
vote
0answers
44 views

What does $SL_2(\mathbb{F}_q)/\{I,-I\}$ look like?

I am not sure what the elements of $$SL_2(\mathbb{F}_q)\big/\left\{ \begin{pmatrix}1&0\\0&1\end{pmatrix},\begin{pmatrix}-1&0\\0&-1\end{pmatrix}\right\}$$ look like for an arbitrary ...
-1
votes
0answers
22 views

Stabilisers, Orbits and Group Theory Help?

a) Draw a regular pentagon with vertices ${v_{1} ,...,v_{5}} \subset \mathbf{R}^2$ such that $v_{1}$ has coordinates (1,0) and the origin in the centre of the pentagon. For each reflection symmetry ...
2
votes
0answers
75 views

Is this a better way to think about Groups as Categories?

I asked a bit ago how to reconcile the category theoretic and set theoretic definitions of groups (groupoid which is a monoid vs the set theoretic definition), and I got the answer I was looking ...
2
votes
2answers
24 views

Prove Zs, Gs (the group of symmetries of the square) and the quaternion group Q are not pairwise isomorphic.

Prove $Zs, Gs$ (the group of symmetries of the square) and the quaternion group $Q$ are not pairwise isomorphic. How would you go about proving. Seems quite difficult. I know that none of the latter ...
1
vote
1answer
38 views

Finding the order of the set of elements fixed by all elements of a group.

I've got an old exam question that I can't figure out how to solve. If anyone could let me know what theorems and lemma's I might find useful, please let me know. Let G be a group of order $p^m$ for ...
3
votes
3answers
72 views

Is every normal subgroup of a finitely generated free group a normal closure of a finite set?

Let $G$ be a group, $S\subset G$ a subset, then the smallest normal subgroup of $G$ which contains $S$ is called the normal closure of $S$, and denoted by $S^G$. My question is, if $G$ is a free ...
0
votes
2answers
25 views

A peculiar decomposition of elements in a group

Let $G$ be an abelian group. Suppose there exists $n\in \mathbb N$ such that $\forall x\in G, x^n=e$. Let $a,b \in \mathbb N$ such $ab=n$ and $\gcd(a,b)=1$. Let $G_a=\{x^a \; | x\in G\}$ and ...
1
vote
2answers
33 views

Uniqueness of a subgroup of a given order

Let $G$ be a cyclic group with order $n$. Prove that for every divisor $d$ of $n$ there is a unique subgroup with order $d$. For the existence, let $x$ be a generator of $G$. It is easy to check ...
1
vote
3answers
28 views

Finding conjugacy classes

I've been having problems with finding conjugacy classes. I don't really understand how to do it properly. Say we look at a S3 group: $S_3=]e, (12), (13), (23), (123), (132)]$ If we look at just ...
1
vote
2answers
62 views

Is this enough to prove that the group is isomorphic to $S_3$?

I have a relatively complicated group, I will not go into detail about what it is, it a group of automorphisms, and the group-relation is composition, so it is kind of complicated. However, I am ...
6
votes
2answers
56 views

If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.

If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$. I want to use induction to prove this: It is trivial when ...
0
votes
0answers
36 views

Size of conjugacy classes of alternating group $A_{22}$

Let $p,q\in\{13,17,19\}$ and $G=A_{22}$, is it true that for every $x\in G$ we have $(|x^G|,pq)\neq 1$? why?
1
vote
0answers
28 views

On finite groups with a special property on proper subgroups [closed]

Let $G$ be a finite group such that all proper subgroups of $G$ are nilpotent. Then $G$ is solvable.
0
votes
1answer
17 views

How to create lattice diagrame in maple 14?

I am studying lattice diagrame of subgroups of groups and I have already posted one query over here. Now my present query is: I am using MAPLE 14. Can anyone suggest me how to create lattice ...
0
votes
1answer
22 views

What does $C_2^8, C_2^4$ etc in lattice diagrame of subgroup represent?

I am studying lattice diagrame of subgroups of groups. and I came to know about the lattices of $C_4\times C_2$ and $C_8\times C_2$ over here and here. But the problem is: I am unable to understand ...
0
votes
1answer
33 views

What do you need to perform Karatsuba multiplication?

Karatsuba multiplication is usually defined in $\mathbb{N} \times \mathbb{N}$ and computes $$(aB^m+b)(cB^m+d)=acB^{2m} +[(a+b)(c+d)-ac-bd]B^m+bd$$ (where B is the base, usually 10) in only three ...
1
vote
1answer
35 views

Composition of groups

Let's say we have a system of interacting particles that can divided into two populations. The symmetry group of each population is $G$, and the two populations are identical, so that I can exchange ...
-1
votes
1answer
21 views

On the dicyclic group of order $4n$ and the dihedral group of order $2n$ [closed]

Let $Q_{4n}$ be the dicyclic group of order $4n, n\geq 2$ and $D_{2n}$ to be the dihedral group of order $2n$. Then prove that $$\dfrac{Q_{4n}}{Z(Q_{4n})}\cong D_{2n},$$ Where $Z(G)$ is the center ...
0
votes
0answers
22 views

Finitely generated subgroups of $\mathbb Q$ [duplicate]

Is it true that any finitely generated subgroup of $\mathbb Q$ is infinite cyclic? My try: if $I=<a_1/b_1,...,a_n/b_n>$ is a f.g. $\mathbb Z$-submodule of $\mathbb Q$ then all $a_i$'s lie in ...
4
votes
1answer
84 views

group of diffeomorphisms of interval is perfect

Every element in $\mathrm{Diff}([0,1])$, group of diffeomorphisms of interval fixing the endpoints, can be written as a product of commutators since this group is perfect (I don't know the proof ...
0
votes
1answer
26 views

Action on finite non-abelian group

Let $G$ be a finite, non-abelian group. Show that if $Aut(G)$ acts on $G$ by $\sigma.g=\sigma(g)$ for each $\sigma \in Aut(G)$, $g \in G$, then there exist at least three orbits. I think I could ...
3
votes
1answer
47 views

Group isomorphism and matrices

Let $\mathbb{F}$ be a field. Consider the following three groups- $$G=\left\{\begin {pmatrix} 1&a&b\\ 0&1&c\\ 0&0&1\\ \end{pmatrix} : a,b,c\in \mathbb{F}\right\} $$ ...
26
votes
8answers
2k views

Is symmetric group on natural numbers countable?

I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not. I tried to prove it is uncountable somewhat mimicking ...
6
votes
1answer
49 views

Inverses of elements in group algebras

If $G$ is a finite group whose elements are $g_1,\ldots,g_n$ and let $F$ be the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. We define a vector space over $F$ with ...
0
votes
0answers
30 views

Why $N_G(E)/C_G(E)$ has order prime to $p$ where $E$ is a rank 1 elementary $p$ subgroup of $G$?

Let $G$ be a profinite group, with $p$-rank 1. i.e. the largest rank of elementary $p$ sub groups is 1, why $N_G(E)/C_G(E)$ has order prime to $p$ where $E$ is a rank 1 elementary $p$ subgroup of $G$? ...
2
votes
0answers
36 views

Quotient group of $\mathbb{R}^2$ by irrational line

In a section about topological groups, exercise 4.10 in I.M. James' General Topology and Homotopy Theory asks Show that for irrational values of $\alpha$ the factor group of the real plane ...
1
vote
1answer
56 views

On intersection of cyclic subgroups a group

Let $G$ be a group and $A=\{g\in G\mid\langle g,x\rangle\text{ is cyclic for all }x\in G\}$. Why is $A$ a cyclic subgroup of $G$? (Must $G$ be finite?)
0
votes
1answer
50 views

Given adjoint action find original matrix.

Given the Adjoint action of a matrix; $\text{Ad}(g) X_1 = g \, X_1 \, g^{-1} = X_2 $. Where g is in a (matrix) Lie group, $X_1,\; X_2$ are from the Lie algebra, can a $g$ be written in terms of the ...
1
vote
3answers
47 views

Showing that $3$ is a generator of the group $(\mathbb{Z}/31\mathbb{Z})^{\times}$

Show that $3$ is a generator of the group $(\mathbb{Z}/31\mathbb{Z})^{\times}$. I have done the following: The order of the group is $30=2 \cdot 3 \cdot 5$. $3$ is a generator if $3^2, 3^3, ...
3
votes
2answers
36 views

Commutative Monoid and Invertible Elements

I am looking for interesting (naturally occurring?) examples of commutative monoids with "lots" of invertible elements and "lots" of non-invertible elements. An easy way to get examples is use the ...
7
votes
1answer
54 views

Free subgroup of linear groups

Suppose $a,b \in GL(n,\mathbb{C})$, and $\langle a,b\rangle$ is a free group of rank $2$. Is there a way to choose a $c$ to guarantee that $\langle a,b,c\rangle$ is a free group of rank $3$?
-1
votes
0answers
45 views

about proof of be group about a finite monoid by contradiction

G be a finite monoid. if G has unique idempotent then G is a group hint:by contradiction let G is not a group we must show G has inverse until to reach contradiction hence Imposition is void and G ...
-1
votes
0answers
39 views

Why is $D_8 \sim S_2 \rtimes ( \mathbb{Z}_2)^2$?

I read that the group of symmetries of the $n$-cube is isomorphic to $S_n \rtimes (\mathbb{Z}_2)^n$ and a proof of this would be very helpful for my research. However, I am having a difficult time ...
2
votes
1answer
37 views

Different descriptions of the Baumslag-Solitar groups using affine groups

On page 101 of this paper of Laurent Bartholdi (which is an online documentation of the FR package for GAP which allows GAP to manipulate groups generated by automata) he gives a different description ...
2
votes
1answer
47 views

Prove that all normal subgroup definitions are equivalent.

given $ N<G $ I need to prove that all of the below are equivalent: 1) for each $g \in G$ , $n \in N$ $gng^{-1} \in N $ 2) for each $g \in G$ $gNg^{-1} = N $ 3) for each $g \in G$ $gN = Ng$ ...
1
vote
1answer
26 views

similar to (applying) group isomorphism theorems

$G$ is a topological group, $H$ is a subgroup of $G$, $K$ is a compact subset of $G$, if $G=HK$, then $G/H$ is compact? Is this right? I said this conclusion is similar to group isomorphism ...