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

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Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{\lcm (n,k)} \rangle$

Let $G$ be a group. Let $a$ be an element. Let $n,k$ be pozitive integers. Let $m$ be least common multiple of $n$ and $k$. Prove generator $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle ...
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1answer
9 views

$AGL(V) = V \rtimes GL(V)$ with $GL(V)$ acting from the right

For a vector space $V$, I have constructed $AGL(V) = V \rtimes GL(V)$ as the elements $(v, A) \in V \times GL(V)$ (Cartesian product of sets, not a direct product) with multiplication $(v, A) (w, B) = ...
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about groups of order p^2qr

i need help to understend next theorem (page 148) : https://archive.org/stream/jstor-1986340/1986340#page/n11/mode/2up Is same true for groups of order $p^2q^2r$?
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0answers
23 views

Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
5
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1answer
37 views

Groups of order $p(p+1)$

If I have a group of order $p(p+1)$ with $p+1$ Sylow $p$-subgroups how can I prove that all $p$ non-trivial elements not of order $p$ have prime order?
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38 views

group and subgroups [on hold]

Let G be a group and H a subgroup of G. For any element g ∈ G let gHg−1 = {ghg−1 | h ∈ H}, which is called the g conjugate of H. Prove that gHg−1 is a subgroup of H. May I know how we can prove this ...
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1answer
26 views

How do I prove that this is or isn't isomorphic? [duplicate]

$\mathbb{Z}_2 \times \mathbb{Z}_3 \cong \mathbb{Z}_6$? How can I show that the groups are isomorphic? (Or not?)
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When do Sylow $p$ and Sylow $q$ subgroups commute?

Do $p$-Sylow and $q$-Sylow subgroups commute iff both are unique and thus normal? I know that one direction is true: namely that if the $p$-Sylow subgroup and the $q$-Sylow subgroup are normal in the ...
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isomorphism classes of non abelian p - group [on hold]

Let be p an odd prime. Are all isomorphism classes of groups of order p⁶ isomorphic to a semidirect product? What happen whith groups of order p⁵?
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1answer
24 views

If G is not commutative [on hold]

Edit: Since I did not provide enough detail in my explanation in OP: I have tried to prove this for the general case, but have not come across a suitable proof. I was unsure if I then needed to prove ...
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Conjugacy classes with the same caedinality [on hold]

$H\le G$ is normal, $HaH^{-1}$,$HbH^{-1}$ are two conjugacy classes in $H$, suppose a,b conjugate in G,show $|HaH^{-1}|$=$|HbH^{-1}|$.
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1answer
33 views

Why in this sense this homomorphism is injective?

In this proof:enter link description here Page 19, it gives a construction of outer automorphism of $S_6$,it sends $S_6$ to Perm($S_6$/H)= $S_6$, and by the former proof, it is injective.However, it ...
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votes
3answers
75 views

Confused with Cayley's Theorem in group theory.

Cayley's Theorem: Every group is isomorphic to a group of permutations. $\mathbb Z_6$ is a group and $S_3$ is a permutation, but $\mathbb Z_6$ is not isomorphic to $S_3$. $\mathbb Z_6$ is ...
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2answers
40 views

Group with $p+1$ Sylow $p$-subgroups

Given a group $G$ with $p+1$ Sylow $p$-subgroups, I've deduced that $R = P \cap P'$, where $P, P'$ are Sylow $p$-subgroups, has index $p$ in each of $P, P'$; and that all $p+1$ Sylow $p$-subgroups of ...
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29 views

properties on groups of order $p^2qr$

I read somewhere that if $|G|=p^2qr$, $H\subseteq G: |H|= p^2q$, $p>q>r$ primes, then if only $H$ is maximal subgroup, then $H$ is Abelian. Is this problem correct? Are there any same properties ...
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1answer
18 views

How to describe the quotient group Z x Z / < (4, -6)>

While solving a problem on group theory, I encountered the quotient group Z x Z / < (4, -6)>. Here Z is the integer. At first I thought it is just Z/(4Z) x Z/(6Z). But I was wrong. the quotient ...
5
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1answer
314 views

Is this group finite?

Let $G$ be a sub-group of the invertible real matrices of size $n$ (usually noted $GL_n(\mathbb{R})$), such that $\forall M\in G,M^2=I_n$ Is $G$ finite ?
3
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2answers
24 views

When Verbal Subgroups are propers

Let $w$ be a group-word, and let $G$ be a group. The verbal subgroup $w(G)$ of $G$ determined by $w$ is the subgroup generated by the set consisting of values $w(g_1, \ldots, g_n)$, where $g_1, ...
2
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1answer
22 views

Group theory: counting the number of elements in $\mathbb{Z} _p ^*$

Let $p$ be a prime number. Let $d$ is a divisor of $(p-1)$ Let $G$ be a group of integers $\{1,2,\cdots,p-1\}$ under multiplication modulo $p$. How may one prove that the number of elements $a$ in ...
3
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2answers
22 views

Radicable Groups

A group $G$ is said to be radicable if each element is an $n$th power for every positive integer $n$, ie, $G$ is radicable if the equation $x^n = a$ has a solution in $G$ for every positive integer ...
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1answer
22 views

prove that if $\exists a\in A\space:\space stb(a)\not=\{e\}$ then the action is unfaithful

let $G$ be abelian group which acts on non empty set $A$. prove that if $\exists a\in A\space:\space stb(a)\not=\{e\}$ then the action is unfaithful (the kernel of the action is not trivial). ...
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0answers
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Symmetric groups and transitive action

I am trying to show that for $(x,x_1),(y,y_1)$ there exists $g\in S_n$ such that $gx=y$ and $gx_1=y_1$ where $x$, $x_1$, $y$, $y_1\in \{1,2,3,\dots ,n\}$ and $x\neq x_1$, $y\neq y_1$. Is this claim ...
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1answer
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If $G$ has only 2 non-trivial proper subgroups H, N, then H, N are cyclic subgroup of $G$.

If $G$ has only 2 non-trivial proper subgroups H, N , then H, N are cyclic subgroup of $G$. I searched essentially same problem at If $G$ has only 2 proper, non-trivial subgroups then $G$ ...
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1answer
17 views

Index of intersection of subgroups in group

Let $H$ and $K$ be finite index subgroups of a group $G$ with index $h$ and $k$, respectively. I know that $H\cap K$ is of finite index in $H$ and $K$. Is the index of $H\cap K$ in $H$ bounded by ...
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2answers
23 views

Symmetries of a Polynomial

I was wondering how many symmetries the polynomial $(x_1-x_2)(x_2-x_3)(x_1-x_3)$ has, and what they are. I got four: (i) $(x_1-x_2)(x_2-x_3)(x_1-x_3)$ (ii) $(x_2-x_1)(x_1-x_3)(x_2-x_3)$ (iii) ...
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1answer
28 views

Homomorphic images of a group [on hold]

If we consider $Q_8$ i.e. the Quaternion Group,then how to find the homomorphic images of this group?
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10 views

Using Lattice Isomorphism Theorem

I am working on this for my algebra class and I am stuck at the very end. $\textbf{QUESTION:}$ Let $p$ be a prime and let $G$ be a group of order $p^\alpha$. Prove that $G$ has a subgroup of order ...
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20 views

ring morphism from a group ring to another ring

I've read that if $S$ is a commutative ring, then $Hom_R(R[G],S)=Hom_R(R,S)\times Hom_{Gr}(G,\mathcal U(S))$. I've tried to show this equality but I couldn't. If $\phi: R[G] \to S$ is a ring ...
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1answer
31 views

How should I calculate the cosets of a subgroup of $\mathbb Z\times \mathbb Z?$

I'm trying to find the factor group $\mathbb Z^2/H,$ where $H = \{(5k,3k):k\in\mathbb Z \}.$ Would the coset of $H$ containing $(a,b)$ simply be $\{(5k + a, 3k+b):k\in \mathbb Z\}?$ If so, then how ...
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direct product of three square matrix

Suppose that $I_1$ is a $n_1\times n_1$ identity matrix and $I_2$ is a $n_2\times n_2$ identity matrix, and $H$ is $n\times n$ matrix. If $$ \bar H=I_1\otimes H \otimes I_2, $$ and we regard all the ...
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1answer
32 views

Is there a classification for the generating sets of symmetric group?

Is there a classification for the generating sets of symmetric group? Or, is there an algorithm for checking wheather a subset is a generating set? For example, can $S_7$ be generated by all its ...
3
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1answer
33 views

question on subgroup of compact group

Suppose $G$ is a compact metrizable abelian group, is it true that $G$ has no finite index subgroups iff $G$ is connected? Any reference or help is appreciated. Thanks in advance! Here are my ...
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1answer
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Transitive action on Poincare upper half plane

I am trying to prove that the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ via $$ \left( \begin{array}{ c c } a & b \\ c & d \end{array} \right)z\rightarrow \frac{az+b}{cz+d} $$ ...
3
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1answer
46 views

No simple groups of order 9555: proof

While reading through crazyproject, I came across the following proof that there were no simple groups of order $9555$. However, I do not understand the step that says: "Moreover, since 7 does not ...
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1answer
41 views

Show that $D_n$ is a subgroup of Perm($\mathbb{C}$).

For $ n \in \mathbb{N}$ and $0 \le r <n$, define $f_r : \mathbb{C} \to \mathbb{C}$ ; $z \mapsto ze^{2\pi i r/n}$ and $c: \mathbb{C} \to \mathbb{C}$ ; $z \mapsto \bar{z}$. a) Let $D_n = \{ f_0, ...
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78 views

Union of subgroups is a subgroup if and only if one subgroup is a subset of the other

Let $H$ and $K$ denote two subgroups of a group $G$. Prove that the union $H \cup K$ is a subgroup of $G$ if and only if $H \subset K$ or $K \subset H$.
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Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
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1answer
16 views

How can I prove that the inverse of $n-1$ in $U(n) = \mathbb{Z}_n^{\times}$ is $n-1$?

Where $U(n)$ is multiplicative group $mod(n)$. It seems obvious but how can I actually prove it? From modular arithmetics we have: $(n-1)a = nk+1$, so $a=(nk+1)/(n-1)$, which should be an integer ...
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Group of order 36

I follow a proof which show that any group of order $36$ has a normal Sylow subgroup. The author suppose that $G$ has no normal subgroups of order $9$ or $4$. Then $G$ won't have subgroups of order ...
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A question about a intransitive group [on hold]

Assume that the intransitive group $G$ has degree $n$ and minimal degree $n-1$. If no transitive constituent of $G$ has degree $1$, then they all are faithful and all except one are regular. Any ...
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Find number of elements of order p in a group

Given a group $\mathbb{Z}_{p^2q}$ where $p$ and $q$ are distinct primes, how to find the number of elements of order $p$; and how to be sure whether they exist . ($\mathbb{Z}_{p^2.q}$ is the addition ...
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Translation of an old proof

I have an old paper, Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459. in the Germany language. Is their a way to access a translation ...
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3answers
251 views

Does every homogeneous space allow a group structure?

Let $(X,\tau)$ be a homogeneous space, that is for all $x,y \in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there a group operation $*:X\times X\to X$ such that ...
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1answer
26 views

Is it true that all proper normal subgroups of $D_{24}$ abelian?

Is it true that all proper normal subgroups of $D_{24}$ abelian ? If Yes, is it true only for $D_{4n}$ groups, or for all $D_{2n}$. I was trying to list all proper normal subgroups of $D_{24}$, Using ...
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How can I have a copy of this old paper? [on hold]

How can I have a copy of this old paper and a translation of it? Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459.
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33 views

Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$

Let $p$ be the smallest prime dividing $|G|$, and suppose that some $P \in \mathsf{Syl}_p(G)$ is cyclic. Prove that $N_G(P) = C_G(P)$. So I let $G=p^\alpha m$ $p$ does not divide $m$. P is cyclic, ...
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1answer
23 views

Clarification on proof that all groups of order $< 60$ are solvable

I've manged to prove that all groups of order $< 60$ are solvable, using Burnside's theorem. However, I found an alternate proof here Question about solvable groups It states that: "Note that ...
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1answer
21 views

Number of mutually non isomorphic Abelian groups

Let p and q be distinct primes. How many mutually non-isomorphic Abelian groups are there of order p^2q^4. I think there are 6 of them: p^2q^4 q, qp, q^2p q^2, q^2p^2 p, pq^3 pq, pq^3 q, q^3p^2 in ...
5
votes
1answer
50 views

The behavior of quotient groups under homomorphisms

We're learning normal subgroups, kernels, homomorphisms and isomorphisms in abstract algebra right now. I'm trying to tie the ends together: I know that if $G$ is a group, $N$ a normal subgroup of ...
2
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1answer
41 views

Groups of order $2\cdot 31\cdot 61$.

What are all groups (up to isomorphism) of order $2\cdot 31\cdot 61$? Letting $n_p$ be the number of Sylow $p$-subgroups of such a group, $G$, you can show $n_{31}=1$ using the Sylow theorems ...