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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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 ...
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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 ...
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21 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 ...
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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 ...
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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 ...
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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 ...
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3answers
71 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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35 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?
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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1answer
47 views

Group isomorphism and matrices

Let $\mathbb{F}$ be a field. Consider the following three groups- $$G=\left\{\begin {pmatrix} 1&a&b\\ 0&1&c\\ 0&0&1\\ \end{pmatrix} : a,b,c\in \mathbb{F}\right\} $$ ...
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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
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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 ...
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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 $G$? ...
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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 ...
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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?)
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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 ...
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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, ...
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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
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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 $3$?
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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 ...
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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 ...
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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 ...
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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 $g \in G$ $gN = Ng$ ...
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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 ...
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1answer
46 views

Why is this a multiple of $s$?

Let $G$ a finite cyclic group of ordern $n=q_1^{a_1} \cdots q_t^{a_t}$, $g\in G$. If $g^{\frac{n}{q_j}}\neq 1$ then $q_j^{a_j}\mid ord(g)$. Proof: Let $s=ord(g)$. Then $s$ is a divisor of $n$, ...
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14 views

three questions about syndetic sets

$G$ is a topological group. Definition: A subset $S$ of $G$ is said to be syndetic in $G$ provided that $G=SK$ for some compact subset $K$ of $G$. 1.If $S$ is a syndetic subgroup in $G$, then $G/S$ ...
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30 views

Does the subnormal subgroups always form a lattice?

Let $G$ be a group and $W(G)$ the set of all subnormal subgroups of $G$, partially ordered by inclusion. My question is if $W(G)$ always forms a lattice (but not necessarily a sublattice of $L(G)$). ...
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Show that there's no simple group of order $63$, please check my reasoning

I want to show that there's no group of order $63$ which is simple and would like to know if my simple reasoning is correct. I am irritated because this is an exercise which is supposed to be harder. ...
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The order of group $G$ is odd. Prove the mapping $f:G\to G$ by $f(x) = x^2$ is injective.

The order of group $G$ is odd. Prove the mapping $f:G\to G$ by $f(x) = x^2$ is injective For what it is worth this is what I have tried. Assume $x,y \in G$, $f(x) = f(y)$. We want to show $x = ...
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59 views

Group theory question (group of order $15$) [duplicate]

Let $G$ be a group where $o(G)=15$. If $G$ has only one subgroup of order $3$ and only one of order $5$, prove that $G$ is cyclic. Generalize to $o(G)=pq$, where $p$ and $q$ are primes.
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Continuous homorphisms between topological groups.

Let $G,H$ be topological groups and let $\rho: G \rightarrow H$ be a homomorphism of topological groups. I understand this to mean that $\rho$ preserves group structure and is a map between the ...
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45 views

If $N$ is a nontrivial normal subgroup of a nilpotent group $G$, then $N\cap C(G)\neq \langle e\rangle$

This is an exercise from a book of algebra. If $N$ is a nontrivial normal subgroup of a nilpotent group $G$, then $N\cap C(G)\neq \langle e\rangle$. This is my proof: If $N\cap C(G) = \langle ...
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2answers
84 views

How to prove that a given group is isomorphic to Sym(4)?

Given a specific group with 24 elements, I want to prove that it is isomorphic to Sym(4). To begin with, I calculate the orders of my group's elements and they come out as in the order statistics for ...
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2answers
40 views

Let $G$ be a finite group, $ord(G)=p^2$ ($p$ is a prime) prove that there is a subgroup of order $p$ in $G$

Let $G$ be a finite group, $ord(G)=p^2$ ($p$ is a prime) prove that there is a subgroup of order $p$ in $G$ I thout about Sylow theorem but it didn't helped me.
2
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1answer
34 views

Solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$

How to solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$? The origin of this problem is the following question: Show that rank-2 free $\mathbb Z$ module $\mathbb Z^2$ has $p+1$ ...
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1answer
108 views

How to show that the group is abelian?

I have this exercise: a. Let $\sigma \in S_{15}$ be an element of order 5. What type of cycles can occur in the decomposition of $\sigma$ in disjoint cycles? b. Let $S \subseteq S_{15}$ be ...
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1answer
49 views

Let $G $ be abelian group $N,H<G$ prove that $NH $ is a subgroup of $G$ [closed]

Question: Let $G $ be abelian group ,let $N,H<G$ I need to prove that $NH:=\{nh\, ; \,n\in N,n\in H\}$ is a subgroup of $G$.