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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Requesting programming implementations for $\mathbb{F}_{p^n}$ and $SL_2(\mathbb{F}_{p^n})$.

I would like a programming language capable of doing computations over finite fields and matrix groups over those finites fields. I do not want to have to construct bases and what not on my own. What ...
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7 views

Using Peter Weyl theorem to decompose an orbit

Let $G$ be a finite group and let $\pi$ be a unitary representation of $G$ on a Hilbert space $H$. Since $G$ is finite, we have that for every $v \in H$, the orbit of $v$, $\pi (G).v$, is of dimension ...
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Let $ G $ is a finite group and $ K $ is a nilpotent subgroup of $ G $

Let $ G $ is a finite group and $ K $ is a nilpotent subgroup of $ G $. since $ K $ is nilpotent then assume $ K = K_{\pi^{\prime}}K_{\pi} $ so that $ K_{\pi^{\prime}} $ is a $ \pi^{\prime} $-Hall and ...
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2answers
31 views

Is there a name for this property?

Is there a general name for the following properties, (similar to the properties of existence of an additive identity, existence of multiplicative identity etc): For any given set, the intersection ...
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1answer
22 views

Find the order of $\tau^{100}$

Let $\tau= \left( \begin{array}{ccc} ...
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1answer
18 views

Group action of $GL(2, F)$ on the projective line $P(F)$

I refer to section 8.3, page 119 of Algebra, A Computational Introduction by John Scherk. It is about group action of $GL(2, F)$ on the projective line $P(F) = F \cup \{\infty\}$. Given a matrix ...
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3answers
32 views

Cyclic Groups, find the generator

Let $a$ be an element in a group $G$. What is a generator for the subgroup $H = G_1 \cap G_2$ where $G_1, G_2$ are the groups generated by $a^m, a^n$, respectively?
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1answer
23 views

Describing all elements in a factor group using the structure theorem for finitely generated abelian groups

On an exam I took a couple of weeks ago there was this question, which I would like to review as I did not figure it out during the exam. We were given the matrix $$A = \begin{bmatrix} 1 & 2 & ...
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2answers
39 views

Why normal subgroup chains in Galois theory

I have began understanding Galois theory and I had a question regarding the relationships of normal subgroups to field extensions. So given an irreducible polynomial over the rationals $$a_1 + a_2x ...
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1answer
9 views

Freedom in choosing the Cartan subalgebra?

What transformations are allowed regarding the Cartan subalgebra of a given group? Weights for every representation are labelled by the corresponding eigenvalues of the Cartan generators. Therefore ...
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3answers
69 views

Show that $G/H\cong\mathbb{R}^*$

Let $G:= \bigg\{\left( \begin{array}{ccc} a & b \\ 0 & a \\ \end{array} \right)\mid a,b \in \mathbb{R},a\ne 0\bigg\}$ Let $H:= \bigg\{\left( \begin{array}{ccc} 1 & b \\ 0 & 1 \\ ...
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1answer
25 views

Proving property of group-like algebraic structures by means of induction

How do you prove (by means of induction) that the following is true for all group-like algebraic structures? $$\operatorname{ord}(a_1 \circ a_2 \circ a_3 \circ \cdots \circ a_{n-1} \circ a_n) = ...
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2answers
31 views

If $G$ is abelian, it has a subgroup in every order of $|G|'s$ divisors? [duplicate]

Assume that $G$ is an abelian group, I read somewhere that it can be derived from Lagrange's theorem that it has a number of subgroups that is equal to the number of $G$'s divisors. Why does it hold? ...
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2answers
34 views

Finding subgroups of $\mathbb{Z}_{13}^*$

I need to find all nontrivial subgroups of $G:=\mathbb{Z}_{13}^*$ (with multiplication without zero) My attempt: $G$ is cyclic so the order of subgroup of $G$ must be $2,3,4,6$ Now to look for ...
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0answers
12 views

Product of a Householder transformation and reflection through the origin in 3 dimensions

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$ $v^T\cdot w=0$, and the Householder transformation ...
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1answer
29 views

$G$ finite, $P < G$ a Sylow p-subgroup, $N_{G}(P)$ the normalizer is contained in $H < G$, show $N_{G}(H)=H$

Let $G$ be a finite group and $P<G$ be a Sylow p-subgroup. Let $N_{G}(P)$ be the normalizer of $P$ in $G$. Let $H<G$ be a subgroup containing $N_{G}(P)$. Prove that $N_{G}(H)=H$. I've been ...
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2answers
15 views

Proof that If G = (V , E ) is a graph, then S is an independent set ⇐⇒ V − S is a vertex cover.

I have the following : Proof: Proof. ⇒ Suppose $S$ is an independent set, and let $ e = (u, v )$ be some edge. Only one of $u, v$ can be in S . Hence, at least one of $u, v$ is in $V − S$ . So, $V ...
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1answer
25 views

Group theory exercise from “Rational Points on Elliptic Curves”

Let $A$ be an abelian group and for every $m \geq 1$, let $A_m$ be the set of elements $P$ of $A$ such that $mP=0$. Now, suppose that $A$ has order $M^2$ and that for every integer $m$ dividing ...
2
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0answers
26 views

Determining the group generated by a set of roots?

I have a set of 45 roots and I want to know which group is generated by the corresponding generators. In the set are 5 diagonal (=Cartan) generators $$ (0, 0, 0, 0, 0, 0)_1,(0, 0, 0, 0, 0, 0)_2,(0, ...
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2answers
60 views

Subgroups of $\mathbb{Z}_{13}^*$ using “GAP language”

How can I know the nontrivial subgroups of $\mathbb{Z}_{13}^*$ (with multiplication without zero) using GAP language
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2answers
40 views

Show that $G:=\mathbb{Z}_{13}^*$ is cyclic

I need to prove that $G:=\mathbb{Z}_{13}^*$ (without zero with multipcation)is cyclic My attempt: I tried to check each element in $G$ if it is a generator or not: $$ \begin{align} &1^1=1\mod ...
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1answer
35 views

Reference Request: Subgroup of free abelian group is free abelian

I have the following reference Question, meaning that I search for a reference for the following statement: Let $F$ be a free abelian group of finite rank and $U$ be a subgroup. Then there is a basis ...
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54 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
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10 views

Reference on Malcev completion

I need a reference for learning Malcev completion, its associated group scheme and Lie algebra. Thanks!
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1answer
32 views

Determining the center of the p-Sylow subgroup of $S_p $

My Algebra book says without proof that the center of the p-Sylow subgroup (we will call it P) of $S_p$ is the subgroup itself. Now I can understand that P will be a subset of its center as it is of ...
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2answers
19 views

Find isomorphism for an operation

I was trying to solve this problem, but am having trouble seeing why it is an isomorphism. To map from R* to G, I think that the phi function would be Phi(x)=x/2 but that doesn't work. This phi ...
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8answers
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What does it even mean to say 'preserve structure'? [duplicate]

Could somebody give a concrete example of a group structure being preserved in a isomorphism, et cetera? I always hear this 'preserve structure' thing. Ok, could somebody give me a rigorous definition ...
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3answers
37 views

Showing that a subgroup of an abelian group is normal—is this sufficient?

When asked to show that a subgroup $H$ of the abelian group $G$ is normal, does it suffice to say: first, $H$ is a subgroup, so it contains the identity element of $G$ and inverses $h^\prime$ for ...
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1answer
26 views

Latin square property sufficient?

So I know that for any group table, Every row must contain distinct group elements and the same holds for every column for a group table. And this property is called the Latin Square property. ...
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1answer
42 views

Quantity of elements of order $d$ in $Z_n$, with $d \mid n $

I'm studying for an exam and I can't answer this problem. I'd appreciate a hint. What I've got so far: Let $x$ be an element or order $d$. Then, $x\cdot d \equiv 0 \pmod n \Rightarrow n \mid x\cdot ...
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38 views

Am I correct regarding Aut($Z_n$)

In the following pic- shouldn't it be $\Bbb{Z}_{{p_j}-1}$ instead of $\Bbb{Z}_{p_j}$. I think so because Aut$(Z_{p^n}) \cong Z_{p-1} \oplus \underbrace {(Z_p\oplus Z_p \oplus \dots Z_p)}_{n-1\ ...
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2answers
59 views

Prove that the free group generated by two elements is a coproduct of the integers by itself in Grp

That's a problem from Algebra: Chapter $0$ by P. Aluffi, p.78, ex.5.6. One needs to prove that the group $F(\{x,y\})$ is a coproduct $\mathbb Z*\mathbb Z$ of $\mathbb Z$ by itself in the category ...
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0answers
22 views

Stabilizers and equivariant maps

Suppose $G$ acts on a set $X$ and $Y$ and let $f:X \to Y$ be a equivariant map. We immediately have that $$\text{stab}_{G}(x) \subseteq \text{stab}_{G}(f(x)) $$ as if $g \in \text{stab}_{G}(x)$ then ...
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1answer
36 views

Construction of free abelian group from free group

I am reading Fraleigh's Abstract Albebra recently, and I cannot prove a statement about free abelian group: Let $F[A]$ be a free group generated by set $A$ and $C$ is the commutator subgroup of ...
3
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2answers
43 views

Maximal closed subgroups in algebraic groups

Let $G \leq GL(V)$ be an affine algebraic group, over an algebraically closed field. Say that $M$ is a proper subgroup of $G$ which is maximal among the closed proper subgroups of $G$. Does $M$ have ...
2
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1answer
36 views

Determine the isomorphism class of $\mathbb Z^3 / M$ for the subgroup $M$ of $\mathbb Z^3$generated by $(13,9,2),(29,21,5),(2,2,2)$

The problem seems not so hard. My confusion rise from the statement in the solution above that "This question is equivalent to reducing the matrix via row and column operations". Please see the ...
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0answers
28 views

solution verification: find center of this group

we have a group $H$ = $\begin{cases} A= \begin{bmatrix}1 & a & b\\ 0 & 1 & c\\0 & 0& 1 \end{bmatrix} : a,b,c \in \mathbb{Q} \end{cases}$ then find its center $ Z(H)$. My ...
2
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3answers
69 views

Which of the following statements are correct? [on hold]

T he order of the smallest possible non trivial group containing elements $x$ and y such that $x^7=y^2=e$ and $yx=x^4y$ is (A) $1\space\space$ (B) $2\space\space$ (C) $7\space\space$ (D) $14$
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36 views

how to generate equivalence relations for quotient group and verify them and what input relations is suitable for subgroup [on hold]

1. similar gap system, which function can output coset table ? if find example below, ...
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2answers
25 views

Generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$—are they multiplicative or additive?

When I was asked to find the generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$, which are denoted as $\phi(14)$, I did not realize whether the generators are multiplicative ...
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0answers
53 views

How to compute $\mathbb Z_n \times \mathbb Z^*_m$? [on hold]

How to compute $\mathbb Z_n \times\mathbb Z_m^*$? (Here $\mathbb Z^*_m$ is the unit group mod $m$ and $(m,n)=1$.) In the paper Multiplicative properties of sets of residues it is said that by ...
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3answers
73 views

Looking for help to understand example of Group

I am looking for someone to help me to understand what is going on in the following example, from Hersteins "Topics in Algebra". It says, Let $G$ be the set of all $2*2$ matrices $$\begin{pmatrix} ...
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1answer
32 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
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2answers
36 views

From group isomorphisms to algebra isomorphisms

Let $A$ be an algebra and let $A^{\ast}$ be the subset of units (that is, invertible elements) of $A$. Then $A^{\ast}$ is a group under the multiplication of $A$. Let $f^{\ast}:A^{\ast}\to A^{\ast}$ ...
3
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2answers
24 views

Isomorphism between $H/H\cap N$ and $HN/N$

Suppose we have a group G under operation +, and let H be a subgroup and N a normal subgroup. I want to prove that $H/H\cap N$ is isomorphic to $HN/N$. Where, if I am not mistaken: $H/H\cap N = ...
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0answers
15 views

definition of line complex in projective space

In a paper of "R.H.Dye", which you can find here: http://link.springer.com/article/10.1007%2FBF02413785#page-1, I face with a mathematical object "line complex in projective space PG(2n−1,q), I need ...
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1answer
17 views

Explicitly compute the trace for the tautological representation of $D_4$ of $\mathbb{R}^2$.

Fix a finite dimensional representation $\rho: G \longrightarrow GL(V)$ of $G$. Its trace is defined as the function $tr:G \longrightarrow F$ defined by $tr(g) = tr(\rho(g))$. Explicitly compute ...
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4answers
51 views

Conjugacy Classes of a group G - Intuitive Understanding

How can I intuitively understand conjugacy classes of a group G. I feel I have a strong understanding of Equivalence Relations, and just completed the proof showing that conjugacy is an equivalence ...
2
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0answers
51 views

Upper bound on groups of order 60

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short ...
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4answers
88 views

In need of a group theory textbook.

I am in need of a group theory textbook for a good summer review. I have already studied from various books (mostly "group theory" part from basic algebra books) and the lecture notes of my teacher, ...