The study of symmetry: groups, subgroups, homomorphisms, group actions.

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Does this subgroup of $\mathrm{SL}(2,\mathbb{C})$ have a a name?

The set of matrices $g$ characterized by $g=\begin{pmatrix}a&ib\\ ic&d\end{pmatrix}$, where $a,b,c,d \in \mathbb{R}$ and $ad+bc=1$, can be easily shown to be a subgroup of ...
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61 views

Is it “group axiom” or “group definition”?

Some text books of group theory use "group definitions" when introducing group, and some other text books use "group axioms". But it is obvious that terms "definition" and "axiom" are different. Which ...
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45 views

Show for each $c$, $\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$ is an abelian group under multiplication of congruence classes

Show for each $c$, the set $$\{x + c \Bbb Z \in \Bbb Z/c\Bbb Z\mid x^{c−1} \equiv 1 \pmod c\}$$ is an abelian group under multiplication of congruence classes.
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For a discrete abelian group $G$, is the Gelfand Representation of $\ell^1(G)$ injective?

Given a discrete group $G$, we can consider the Banach $*$-algebra $\ell^1(G)$, with convolution product $(\xi*\eta)(g)=\sum_{h\in G}\xi(h)\eta(h^{-1}g)$ and involution ...
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59 views

Order Preserving Isomorphism

If abelian group G has an archimedean order then there is an order preserving isomorphism $\phi$ of G onto a subgroup of $\mathbb{R}$. Here we can say that G is archimedean totally ordered abelian ...
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58 views

groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
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31 views

representation of extraspecial group

Let $H=\langle x,y\rangle$ be a non abelian group of order $p^3$ which has exponent equal to $p$. Also let $G$ be an extraspecial group of order $p^{2r+1}$ exponent $p$. We know that $G$ is isomorphic ...
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55 views

Improper Rotations in Even Dimensions

In odd dimensions, we can represent any improper "rotation" G as $-\mathbb{1}\cdot R$ where $R\in SO(d)$. In even dimensions, $-\mathbb{1} \in SO(d)$ and we cant do this. Is there a way of writing an ...
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35 views

For $k$ random perms of an $n$-set $\mathrm{Pr}[\sigma_1\cdots\sigma_k=\sigma_k\cdots\sigma_1]\xrightarrow{k\rightarrow\infty}\frac{2}{n!}$?

Q. Fix $n \geq 2$, and choose $k$ random permutations $\sigma_1\sigma_2\cdots\sigma_k \in S_n$ uniformly at random. Is true that ...
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45 views

Nilpotent groups and their automorphisms

As usually I'm trying to understand things. This time is the following Lemma 1.21. I report also some other stuff to help comprehension. Anyway all the stuff is taken from Casolo Question I ...
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37 views

Variations of M,n,k-games

I just read about M,n,k-games and wondered if the following variation (with fixed $k$) has been studied as well and if there exists a name for it: Two players consecutively mark elements of ${\bf Z}$ ...
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49 views

Normally embedded subgroups reducing in a Hall system

A Hall system of $G$ is a set $\Sigma$ of Hall subgroups of $G$ satisfying the following two properties: -For each $\pi$ divisor if $|G|$, $\Sigma$ contains excatly one Hall $\pi$-subgroup $G_{\pi}$. ...
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36 views

“Bypass” Operations and Groups

So I recently stumbled on this (pdf) collection of group theory related Putnam problems. Problem 1978 A-4 defines a "bypass" operation to be a mapping $\circ:S\times S\mapsto S$ such that $$(w\circ ...
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40 views

Exact sequences of $1 \to A \to SO(N) \to B \to 1$, special orthogonal group

Inspired by the nice post and this, apart from SU(N), now I am particularly looking into the exact sequences of SO(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A ...
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40 views

Exact sequences of $1 \to A \to SU(N) \to B \to 1$, special unitary group

Inspired by the nice post, I am particularly looking into the exact sequences of SU(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A \to SU(N) \to B \to 1$$ where ...
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65 views

On subgroups of a finite abelian group

Let $G$ be a finite abelian group such that $G$ is of odd order or decomposition of the sylow 2-subgroups of $G$ there is at least two cyclic direct factors of maximal order. Then prove or disprove ...
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20 views

Non-central proper normal subgroups of unitary groups over fields

Short version: Can someone give an example of an anisotropic Hermitian form over a field such that its corresponding projective unitary group is not simple? Let $F$ be a (commutative, associative, ...
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36 views

Infinite finitely generated $2$-groups

Can somebody give me an example of a finitely generated infinite $2$-group?
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Group of permutations and disjoint cycles.

Two cycle $[i_1,..., i_r]$ and $[j_1,...,j_s]$ are said disjoint if no integer $i_\eta$ is equal to any integer $j_{\mu}$. Prove that a pemutation is equal to a product of disjoint cycles. I was ...
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27 views

Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. Is ...
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25 views

Poincare polynomial of a finite $G$-module with $G$ being a $p$-group

Recently, I've been reading Shatz's book, profinite groups, arithmetic, and geometry. Let $G$ be a finite $p$-group and $A$ a finite $G$-module such that $pA=(0)$. In the proof of Theorem 19 (p.82), ...
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52 views

Pushout and pullback of short exact sequence of groups

I think that there might be some textbooks which introduce the notions of pushout and pullback of a short exact sequence of groups. However, I cannot find any of them. To be precise, for a given ...
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1answer
72 views

Centre of $GL(n,\mathbb{R})$ [duplicate]

Can you help me regarding the Centre of $GL(n,\mathbb{R})$ $?$ It is easy to see that the diagonals are there. what could be the other elements? It may be an useless question but it came to my mind! ...
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27 views

Question on Proof involving direct products and $\pi$-complements

All groups $G$ are finite, for $N \unlhd G$ we set $\overline{G} := G/N$ and for some $U\le G$ we write $\overline U = UN/N$. Also for a set $\pi$ of primes we denote by $O_{\pi}(G)$ the largets ...
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29 views

Every finite group has a composition series

Every finite group has a composition series. The proof of this statement is as follows Proof. If $|G| = 1$ or $G$ is simple, then the result is trivial. Suppose $G$ is not simple, and the result ...
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40 views

Normal Closure, Normal Interior (Core) and Lattice Theory in Groups

In D. J. S. Robinson's, A Course in the Theory of Groups it is written (on page 16): "If $X$ is a nonempty subset of a group $G$, the normal closure of $X$ in $G$ is the intersection of all the ...
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38 views

Presentations that make the Todd-Coxeter algorithm blow up

Consider the presentation defined, for an integer $n > 1$, by $$G_{n} = \langle x, y \mid xy^n = y^{n+1}x, yx^{n+1} = x^{n}y\rangle .$$ The group defined by this presentation is trivial. Is it ...
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33 views

regular unipotent elements

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p\neq 2$ and $q$ elements. We know that if $p$ is not bad for $G$ then $G$ contains $q^l$ regular unipotent elements ...
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24 views

Automorphism group of an abelian p-group

I'd like to know if it's known the structure of the automorphism group of an abelian $p$-group with the minimal condition on subgroups, for some prime number p. I know that if $A$ is an abelian ...
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36 views

Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
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35 views

Center of a $2$-Sylow

Here we go for the last time with the group $G=GL_2(\mathbb F_3)$. We know that it has three $2$-Sylow subgroups. So let $P\in\operatorname{Syl}_2(G)$. I was searching for its center but I don't ...
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Question to Corollary on $S$-semipermutable subgroups

Two subgroup $A, B \le G$ are said to permute if $AB = BA$. A subgroup $H \le G$ is called $S$-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for primes $q$ not dividing ...
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Finitely generated groups without the minimal condition on subgroups

Definition A group $G$ is said to satisfies the minimal condition on subgroups iff every descending chain of subgroups stops after a finite number of steps. I know Novikon and Adjan proved that the ...
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23 views

semidirect product and homomorphisms which doesn't make any sense

In describing a semidirect product I haven't understood if a homomorphism is well-defined It says that if $G$is a semidirect product between K and Q then there is a homomorphism $\theta:Q\rightarrow ...
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26 views

Summand of a subgroup in a torus

Let $\mathbb{T}^n := \mathbb{R}^n / \mathbb{Z}^n$ denote the $n-$dimensional torus. If $K$ is a closed normal subgroup of $\mathbb{T}^n$, then does there exist a subgroup $L$ of $\mathbb{T}^n$ such ...
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34 views

Calculation related to the number of conjugacy classes of the symmetric group

The symmetric group on $n$ elements, $S_n$, can act on itself by conjugation. The orbits of this action are the conjugacy classes corresponding to integer partitions of $n$. If $S_n$ acts on some ...
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kernel of the sum homomorphism

Given a finitely generated free abelian group $G$ and subgroups $K_1, \dots , K_r$ it is possible to describe easily the kernel of the homomorphism $K_1 \oplus \dots \oplus K_r \to G$ given by $(k_1, ...
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62 views

Why is this a group action?

Let $G$ be a group and let $H$ be an infinite cyclic normal subgroup of $G$ of finite index. Let $K$ be the centralizer of $H$ in G, $$K=C_G(H)$$ and suppose that the index of $K$ in $G$ is 2. Let $E$ ...
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37 views

Let $H$ be a subgroup of $G$ such that for all $x \in G-H$ and all $y \in G$ there exist $u \in H$ such that $y^{-1}xy=u^{-1}xu$ …

Let $G$ be a group and $H$ be a subgroup of $G$ such that for each $x \in G-H $ and each $y \in G$, there is a $u \in H$ that $y^{-1}xy = u^{-1}xu$. Prove that $H \lhd G$, and $G/H$ is abelian. I saw ...
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40 views

If $(ab)^{3}=a^{3}b^{3}$ and $(ab)^{5}=a^{5}b^{5}$ for all $a,b \in G$ then $G$ is Abelian? [duplicate]

How does $(ab)^{3}=a^{3}b^{3}$ and $(ab)^{5}=a^{5}b^{5}$ for all $a,b \in G$ implies that $G$ is abelian? I know that the first criteria alone isn't enough because there's a counterexample. What could ...
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Algebraic extension of an abelian group.

Let $H \leqslant G$ be abelian groups. Suppose there were $k \gt 1$, $x \in G \setminus H$, such that $x^k = b \in H$. Then Define $H(x) = \{h x^n : n \in \Bbb{Z}, h \in H\}$ to be a simple ...
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Calculate all the local automorphisms

The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$ How to compute all the ...
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34 views

Subgroups of finite abelian group

Let $H$ be a subgroup of a finite abelian group $G$ , then how to show that $G$ has a subgroup which is isomorphic to $G/H$ ? N.B. This is not a duplicate of Subgroups of a finite abelian group , ...
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72 views

maximal irreducble subgroups of $SL(2,q)$

If $H$ is a maximal solvable irreducible subgroup of $GL(2,q)$ then intersection $H \cap SL(2,q)$ is maximal solvable irreducible subgroup of $SL(2,q)$. Why is it true? Maybe this is not true, but ...
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23 views

Centralizing a maximal flag in a symplectic group

Short version: I'm confused about maximal totally-isotropic flags versus maximal flags: do they have the same centralizer in the classical group? Let $F$ be a field, $V$ be a finite dimensional ...
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34 views

In HNN Extensions, is the stable letter assumed to have order 2?

I have been reading about HNN extensions, and trying to understand their construction with regard to normal/reduced form. First, the construction of an HNN extension as I understand it is given, ...
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Is my proof correct? If $A \trianglelefteq G$ is abelian, and $B \leq G$, then $A \cap B \trianglelefteq AB$.

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: If $A$ is an abelian group with $A \trianglelefteq G$ and $B$ is any subgroup of $G$ prove that $A \cap B ...
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48 views

Number of homomorphic images of a cyclic group $G$ of order $n$ is the number of divisors of $n$?

Prove that the Number of homomorphic images of a cyclic group $G$ of order $n$ is the number of divisors of $n$ Attempt: If $\phi : G \mapsto G'$ is a homomorphism, then if $\phi(g) = g'$ where $g ...
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28 views

How can i give this isomorphism between a Galois group and $S_3$?

I have shown that if $x$ is the nontrivial cubic root of the unity and that if $y$ is the real cubic root of $2$, then $Q(x,y)$ is a Galois extension whose Galois group has order $6$. I know that the ...
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61 views

Isomorphism of semidirect products

Suppose we have $\varphi,\psi : H \to Aut(N)$, so that $\varphi = Ad_g \circ \psi$ where $Ad_g:N\to N$ by $n\mapsto gng^{-1}$ for some $g \in N$. Prove that $N \rtimes_{\varphi} H \backsimeq N ...