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

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
29 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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8 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
23 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 ...
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19 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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29 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
2
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2answers
38 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
34 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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52 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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0answers
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
31 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
998 views

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
36 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
56 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
18 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
35 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
39 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 ...
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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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26 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
68 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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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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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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52 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
31 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 $f^{\ast}:A^{\ast}\to A^{\ast}$ ...
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2answers
23 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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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 ...
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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
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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, ...
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0answers
38 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that ...
2
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0answers
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Why is the trivial group a zero object for the category of groups, but the empty set isn't a zero object for the category of sets? [duplicate]

I understand that the zero ring can't be a zero object for the category of rings, because in that case the 'arrows' are ring homomorphisms which, by definition, but maintain the unit. But in the ...
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A qualifying-exam problem related to Clifford theorem in representation theory

I am not sure : does this problem can be solve directly by Maschke's theorem, which states that every representation of a finite group is completely reducible. Perhaps I made some stupid ...
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1answer
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Exercise from Serre's “Trees” - prove that a given group is trivial

In Serre's book "Trees" on page 10 the following exercise is given: Show that the group defined by the presentation $$x_2x_1x_2^{-1}=x_1^2, \hspace{7pt} x_3x_2x_3^{-1}=x_2^2, \hspace{7pt} ...
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0answers
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How do roots act on weights?

In Lie theory it's possible to compute things very explicit using tensor methods. For example, we can use an explicit matrix for each generator $T^a$ and compute the "action" of this generator on an ...
3
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1answer
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Why can we assume $N$ to be a $p$- group?

Let $G$ be a finite solvable group such that if three distinct primes $p,q$ and $r$ divides $|G|$ then $G$ does not contain any element of order the product of two primes and $G$ is minimal w.r.t ...
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1answer
46 views

Are there any examples of non-abelian subgroups of abelian group? [on hold]

Are there any examples of non-abelian subgroups of abelian group?
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1answer
79 views

Group as a $\mathbb Q$-vector space

Let $G$ be a torsion free abelian group of having $n$ number of maximally rationally independent elements $r_{1}, r_{2}, ..., r_{n}$ and assume that $G$ is not finitely generated. Is this correct ...
2
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1answer
44 views

Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
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1answer
26 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with ...
2
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1answer
52 views

Looking for various proofs there is no faithful representation $\rho:S_4\rightarrow GL(\mathbb{R}^2)$

I stumbled on the question when thinking about a representation $\rho:D_4\rightarrow GL(\mathbb{R}^2)$ as symmetries of the four points $\{(\pm 1, \pm 1)\}$. There didn't seem to be a good geometric ...
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3answers
71 views

Find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $.

I'd like to find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $. What I've tried so far: I tried to do $ \varphi(Id) = \bar{0} = \bar{4} $ (as somebody used here). But then I realized ...