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Questions tagged [group-theory]

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 is the study of groups.

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If $G$ is nilpotent of class 2 and if $a \in G$, then the function $\alpha: G \rightarrow G$ …

If $G$ is nilpotent of class 2 and if $a \in G$, then the function $\alpha: G \rightarrow G$ defined by $x \rightarrow [a,x]$ is a homomorphism. Proof: Let $x,y \in G$ Then $\alpha(xy) = [a,xy] = [a,...
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Regarding the Intersection of $p$-Sylow Subgroups

The Problem: Let $G$ be a finite group and $p$ a prime divisor of $|G|$. Prove that a normal $p$-subgroup of $G$ is contained in every $p$-Sylow subgroup of $G$. My Attempt(s): Here's a slight ...
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3answers
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“Distinguished element” in group theory

I have asked that given a short list of triples of a set, a binary operation on the set, and a distinguished element, whether they forms a group. For example, the set is Z, the operation is addition ...
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1answer
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Characterization of anti-homomorphisms

Let $G$ be a group and $G^{op}$ denotes its opposite group. It is well-known that the functor $F$ from $Grp$ to itself, defined by $$ \begin{aligned} G&\mapsto G^{op}\\ x&\mapsto x^{-1}\\ \...
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Normalizer in matrix groups

I have the problem of calculating the normalizer of $\begin{bmatrix}      \lambda & 0 \\ 0 & \lambda^{- 1} \end{bmatrix} $ in the group $\begin{bmatrix}      \cos (\theta) & -\sin (\theta) ...
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How to understand the structure of the interesting graph obtained from the group?

Let $G = A_5$ and $H < G$ is subgroup of $A_5$ generated by $(12)(34)$, and $(125)$. Define graph $\Gamma$ by a vertex set is element of $G$ and elements $x$ and $y$ adjacent if $|H^x \cap H^y| =...
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Proof of Leibniz formula for determinants [on hold]

im trying to understand the following proof but it is not very clear for me, can someone walk me through it? Proof Thank you in advance
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Proof verification for order of normalizer of subgroup of symmetric group

I'm studying for a qualifying exam in algebra and my abstract algebra has gotten quite rusty, so I'm looking for a proof verification. The problem is as follows: Let $S_7$ be the symmetric group of ...
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1answer
22 views

Basic theorem for solvable groups not true for nilpotent groups - counterexample.

it's my first question on MathStackExchange so please be tolerant. Let H be a normal subgroup of group G. If H and G/H are both solvable, then G is solvable. But H nilpotent and G/H nilpotent doesn't ...
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14 views

Two versions of class equation

I have seen two versions of class equation. If $\bar{x}_1$, ..., $\bar{x}_m$ where $x \in G$ are distinct conjugacy classes of G, then $|G|= \displaystyle\sum_{n=1}^{m} [G:C_G(x_i)]$. Sometimes it ...
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Embedding of $G$ naturally in symmetric group and its normalizer

Let $G$ be a finite group of order $m$, and for each $g\in G$, let $L_g:G\rightarrow G$ be the permutation $x\mapsto gx$. Then $\{L_g:g\in G\}$ is a subgroup of $S_m$ isomorphic to $G$. Consider any ...
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Congruence of integers and primitive roots

Lemma: In $\mathbb{Z}$, if $l\ge 1$, $p$ any prime, and $x\equiv y\pmod{p^l}$ then $x^p\equiv y^p\pmod{p^{l+1}}$. The proof is by Binomial theorem. Assume the Lemma. Suppose $x^{p^l}\equiv 1\pmod{...
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$p$-sylow subgroups of $SL(3, \mathbb {Z}_p)$

I wonder how many $p$-sylow subgroups of $SL(3, \mathbb {Z}_p)$ are there. ($p$ is any prime) Rather than finding generators, I used the fact that |$GL(3, \mathbb {Z}_p)$| = $({p}^3 - 1)({p}^3 - p)({...
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Dihedral and permutation groups isomorphism [on hold]

Why dihedral group $\mathbb{D_4}$ is isomorphic with $\mathbb{S_4}$ and not isomorphic with $\mathbb{S_8}$?
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Is the symmetry group of the square isomorphic to $\mathbb{Z_8}$? to $\mathbb{S_8}$? to a subgroup of $\mathbb{S}_8$? [on hold]

Exercise: Mark True or False. Explain why. a) The symmetry group of a square is isomorphic to $\mathbb{Z_8}$. b) The symmetry group of a square is isomorphic to $\mathbb{S_8}$. c)The ...
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Group action on convex cone

I was wondering if I could get some help understanding the following fact in an academic paper. The setup is as follows: An open subset of $\Omega \subset \mathbb R^k$ is an open convex cone if it ...
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Center of Carnot Group [on hold]

I know the center of the Heisenberg group (over $\mathbb{R}$) is isomorphic to $\mathbb{R}$, but what about the wider class of Carnot groups? Are their centers also $\mathbb{R}$?
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1answer
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Finding a set on which a group acts on

I have a given group, defined by its table, namely: Now I am asked to find a set $X$ where this group acts upon non trivially. I have problems understanding this question. As I understand, I am ...
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1answer
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Proving $S_3$ is not a coproduct of cyclic groups

Aluffi II.6.11 suggests proving the above. Here's the sketch of my proof by contradiction. Assume that $S_3$ is a coproduct of a family $\mathcal{C}$ of cyclic groups $C^i$. Using the universal ...
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1answer
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Elementary question on quotient group construction

I am going though an introductory proof text book and was hoping if someone could verify/correct my attempt at the following problem : Let $(\Bbb{Z}[\sqrt3],+,.)$ where $\Bbb{Z}[\sqrt3] \subset\Bbb{...
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An exercise in an introduction to the theory of groups [on hold]

4.4 Let $G$ be a finite p-group; show that if $H$ is a normal subgroup of G having order $p$, then $H$ is a subgroup of $Z(G)$. $Z(G)=\left\{x\in G| xy=yx, \forall y\in G \right\}.$ Can you help me ...
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Group Theory - Natural Coordinate Function

I am studying Shattered Symmetry - From the Periodic Table to the Eightfold Way; in order to start getting into group theory, and quantum mechanics. I am struggling to understand what the author ...
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1answer
65 views

Sum of subsets is the whole set?

Let $A$ be subset of $\mathbb{Z}_n$. Define $A \oplus A = {\{a \oplus a’ : a \in A, a’ \in A}\}$, where $a \oplus a’ = a+a \mod n$. If $A$ is not contained in a coset of a proper additive ...
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Quadrants of a cyclically ordered group

I am trying to prove that if there are two different positive elements in a non-linearly cyclically ordered group, then every quadrant of the group is not empty. Is this a correct statement? ...
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1answer
51 views

Doubt regarding “Elementary approach to proving that a group of order 9 is Abelian”

I am trying to understand the solution of this problem . I am unable to understand why : If $yx=x^2y$, then $yxy^{-1}=x^2$. This means that $y^3xy^{-3}=x^8 $ It seems like I am missing something ...
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0answers
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Lattices are not solvable in non-compact semisimple Lie groups

I'm trying to prove the following result. If $G$ is a non compact semisimple Lie group (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is not solvable, and $[\...
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1answer
51 views

Why is $\mathbb{Q}[i]$ not a subgroup of $\mathbb{R}$?

$i$ is the imaginary unit on the complex plane. I am confused on a particular portion of the definition of a subgroup. Let $G$ be a group and $H$ is a subset of $G$. A part of the definition is ...
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1answer
50 views

The number of group homomorphisms from $\mathbb{Z_2}\times \mathbb{Z_2}\rightarrow \mathbb{Z}_4$

Find the number of group homomorphisms from $G_1:=\mathbb{Z_2}\times \mathbb{Z_2}\rightarrow G_2:=\mathbb{Z}_4$ Let $a,b$ be the generators of $G_1$. Then $0=f(0)=2f(0)$ so we have $f(a)\in \{0,2\}$ ...
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1answer
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$|G|=p^2q^2$ where p,q are primes, $p<q$ and $p \nshortmid (q^2-1)$ Show $G$ is Abelian.

$|G|=p^2q^2$ where p,q are primes, $p<q$ and $p \nshortmid (q^2-1)$ Show $G$ is Abelian. ($p$ does not divide $q^2-1$) The first thing to note is that $(q^2-1)=(q-1)(q+1)$ so we can conclude that ...
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Infinite abelian subgroup of $SL(2,\mathbb{C})$ [on hold]

Are there any infinite abelian subgroups of the special linear group $SL(2,\mathbb{C})$?
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What is $\ell^2(\Gamma)$ for a discrete group $\Gamma$?

I am trying to get my head around the left regular representation of a group, and I am not sure of the definition of the space $\ell^2(\Gamma)$ if $\Gamma$ is a discrete group. To quote what I am ...
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1answer
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Nonabelian group of order $p^n$ for every $p$ prime and $n \in \mathbb{N}$

Nonabelian group of order $p^n$ for every $p$ prime and $n \in \mathbb{N}$ My idea was to just create some sort of generalized dihedral group, that's what I called it at least, maybe it is. $G = <...
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Is there a name for a group describing universal symmetry? [on hold]

I am looking for the name of a group which is the opposite of the trivial group. The trivial group is a group describing the symmetries of a transformation where only the identity transformation ...
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2answers
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Help with homomorphism

Can someone please explain where the equation on the RHS of ⇒ came from in the first line of the maths solution? Question: A group G is called simple if it is not the trivial group and its only ...
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For a group morphism $G \rightarrow H$, give the image of the generators.

I have two group tables, both with order $24$. Now I have to write the images of the first generator as the second generator. The group elements are all letters, thus $G=\{A,B,C,...,V,W,X\}$ and the ...
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1answer
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What is the radical of upper triangular matrices?

Let $B$ be the algebraic group of upper triangular matrices with entires in some algebraically closed field. I would like to know what is the radical of this group is... Any explanation would be ...
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What if the sum in the RHS of the Class Equation is extended to the whole $G$?

For a finite group $G$ of order $n$, the Class Equation reads: $$n=\sum_{b_j \in B}[G:C_G(b_j)]$$ where $B$ is a set of representatives of the conjugacy classes of $G$. Q: Can else come from ...
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Embedding a polynomial ring into $\mathbb{Z}_{n^s}$

My setting is the following: $n$ is a product of two big primes (RSA-like), and I am given a $R = \mathbb{Z}/n^s\mathbb{Z}$ as a space to work with. I would like to represent elements of $R$ as ...
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3answers
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Prove that every group of order 15 is abelian? [duplicate]

I had seen this proof at many places, but everywhere sylows theorem is used. So is their any way to solve it without using sylows theorem?
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1answer
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Axiomatisable Groups

Let A be a set of sentences (“proper axioms”) in a first-order language L with equality. Let us write Mod(A) for the class of all models of A which respect equality. We say that a class of L-...
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A dodecahedron out of five tetrahedra AKA Partitioning an orbit in X/K into orbits under cosets of H in K

Consider the orbit space $X/K$ with $X$ a symmetric space and $K$ a group. Let $x$ represent an orbit $Kx$ in $X/K$. Now let's introduce a subgroup $H \subset K$, split up $K$ into cosets $aH$ (with $...
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embedding a proper normal subgroup of a p-group

I am trying to prove this claim: Let $G$ be a finite $p$-group and $H$ be a proper normal subgroup of order $p^k$,then $H$ can be embedded into a normal subgroup of order $p^{k+1}$. Here is my ...
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1answer
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How to prove $Q_8$ is not subgroup of $S_4$ by sylow thm?

Prove $Q_8$ is not subgroup of $S_4$ Hi. It is trivial that $D_4$ is a subgroup of $S_4$. But the case $Q_8$ make me confuse why this group is not a subgroup of the $S_4$. Why is the $Q_8$ isn't ...
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1answer
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$H^1(SO(2),T(2))$, or finding all subgroups of the group of motions of the plane which are complementary to the translation subgroup

My question is motivated by this part of 'Basic Notions of Algebra' from Igor R. Shafarevich. He mentions the group of motions of the plane in the following context. Let denote $SE(2)$ the 2-...
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Raising to the power of $i$

We all know the usual exponentiation $a^i$ in the complex setting; one can ask is there such a map in a given group; more specifically is there a criterion in the general sense such that given an ...
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2answers
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Inductive Proof of Group with Prime Decomposition is Isomorphic to Direct Product of Cyclic Groups

My lecturer set as a bonus exercise the following induction proof: If $G$ is a finite abelian group $|G| = p_1^{n_1} \cdots p_s^{n_s}$ is the decomposition of $|G|$ into a product of distinct prime ...
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Group structure on the set of maps in a model category

Let $B$ be an object equipped with two maps $ \mu: B \times B \to B$ and $\rho : B \to B$ in a cocomplete model category $\mathcal{M}$ such that $(B, \mu, \rho )$ is a group object. These two maps ...
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0answers
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Generators of the second quotient of the lower central series

Let $F$ be a free group on $S=\{ x_{1},x_{2},\dots,x_{n}\}$ and let $y_{i}=x_{i}F'$ be the coset classes of $x_{i}$, where $F'=[F,F]$ the derived subgroup. Is the set of commutators among $y_{i}$ a ...
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1answer
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Proving every subgroup of a nilpotent group is subnormal

Let $G$ be a nilpotent group, i.e. $\exists$ an upper central series: $1\triangleleft Z_1(G)\triangleleft Z_2(G)\triangleleft ....\triangleleft Z_k(G)=G$ Where $\frac{Z_{i+1}}{Z_i}=Z(\frac{G}{Z_i})$ ...
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1answer
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Composition of elements in dihedral group

I've come across the following example: $$ρ^3·σρ^2 = ρ^2σρ^{−1}ρ^2 = ρ^2σρ = ρσρ^{-1}ρ = ρσ = σρ^{−1} = σρ^5$$ And was wondering if it is true in general that $ρ^i·σρ^j = σρ^{i+j}$? I know that $ρσ =...