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

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2
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10 views

Characterizing $\text{PGL}_2(\mathbb F_p)$

Where can I find a description and proof of the basic structure of $\operatorname{PGL}_2(\mathbb{F}_p)$ (Number of elements with each order, conjugacy classes, etc.) which is understandable by an ...
0
votes
0answers
14 views

Centralizer of element in group PSL(2,F_p)

Is it true, that $\forall g\in PSL(2,F_p)\setminus\{e\}$, $Z(g)$ is Abelian? I think that this is true, but i can't find simple prove.
1
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2answers
22 views

what is the maximum order one element could have in permutation group $S_5$?

what is the maximum order one element could have in permutation group $S_5$? Tried: $|S_5| = 5!=2^3\times3\times5$ but I don't think it has anything to do with the cardinality as often seen in ...
3
votes
3answers
58 views

How to prove if $G$ is a group with every non-identity element having order 2 and $H$ is a subgroup, $G/H$ is isomorphic to a subgroup of $G$.

This isn't a homework problem. I'm preparing for an exam, and I have no idea how to solve this problem. Let $G$ is a group such that every non-identity element has order $2$. Let $H$ be a ...
3
votes
1answer
44 views

Proving that the multiplicative group mod p (p is prime) is cyclic

Let $\mathbb{Z}_p^*$ be the group of integers $\{1,2,3,\dots,p-1\}$ under multiplication mod $p$, where $p$ is a prime. Given the following two facts: If $d$ divides $p-1$, then there are exactly ...
0
votes
1answer
24 views

$H$ is the smallest subgroup of $S_n$ containing the transposition $(1,2)$ and the cycle $(1,2,…,n)$ [on hold]

For $n>2$, if $H$ is the smallest subgroup of $S_n$ containing the transposition $(1,2)$ and the cycle $(1,2,...,n)$ , then (A) $H = S_n$ (B) $H$ is abelian (C) The index of $H$ in $S_n$ ...
5
votes
2answers
31 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
3
votes
1answer
37 views

Group objects in category of $\mathcal{Set}$ are groups - How to proof?

Reading about group objects in categories, it's a fact that a group object is in the category of $\mathcal{Set}$ just a common group. I am trying to give an actual proof of this, but I'm a bit ...
3
votes
2answers
40 views

$H,H'$ normal in $G,G'$ respectively and $G \cong G' $ and $H \cong H' $ $\implies$ $G/H \cong G'/H'$ ? [duplicate]

Let $H$ be a normal subgroup of $G$ and $H'$ be a normal subgroup of $G'$ such that $G \cong G' $ and $H \cong H' $ , then is it true that $G/H \cong G'/H'$ ? Denoting by $f$ the isomorphism between ...
0
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0answers
50 views

Wielandt's Exercise [on hold]

Wielandt, Exercise 5.2. 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 ...
2
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0answers
13 views

Do locally soluble and the maximal condition on subnormal subgroups imply soluble?

Here is an interesting question. Let $G$ be a locally soluble group. We know that if $G$ satisfies $Max$ (the maximal condition on subgroups), then $G$ is trivially soluble. On the other hand, if $G$ ...
7
votes
2answers
238 views

Words in the Category of Sets

I was wondering about free objects in different categories and the "words" in those categories. I think I have a generally good grasp on the idea, but I started to think about stranger free objects ...
0
votes
0answers
9 views

Is there a relationship between orthomorphisms of groups and orthomorphisms on Riesz spaces ?

Is there a relationship between orthomorphisms of groups and orthomorphisms on Riesz spaces ?
1
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3answers
37 views

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 $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{m} ...
1
vote
1answer
20 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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0answers
25 views

about groups of order p^2qr [on hold]

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 $p^2q^2r$?
3
votes
0answers
31 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 ...
6
votes
1answer
46 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?
0
votes
0answers
40 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 ...
1
vote
1answer
30 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?)
4
votes
1answer
35 views

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 ...
-3
votes
0answers
13 views

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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votes
1answer
26 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 ...
-1
votes
0answers
19 views

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
34 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 ...
0
votes
3answers
81 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 ...
4
votes
2answers
48 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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0answers
28 views

finite solvable group with a certain property

Let $G$ be a finite solvable group and for each proper normal subgroup $N$ of $G$, $\frac{G}{G^{\prime}N}\cong \Bbb{Z}_p\times\Bbb{Z}_p$ or $\Bbb{Z}_{p^n}$, where $n\geq 1$, $p$ is a prime number ...
0
votes
1answer
33 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 ...
1
vote
1answer
19 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 ...
6
votes
1answer
319 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
votes
2answers
28 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
votes
1answer
23 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
24 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 ...
1
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1answer
24 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). ...
0
votes
0answers
14 views

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 ...
0
votes
1answer
21 views

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 $G$ ...
0
votes
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 ...
0
votes
2answers
25 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) ...
0
votes
1answer
29 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?
0
votes
1answer
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 ...
1
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1answer
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 ...
1
vote
1answer
32 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 ...
0
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0answers
17 views

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 ...
4
votes
1answer
36 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
votes
1answer
35 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 ...
3
votes
1answer
14 views

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
votes
1answer
47 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 ...
2
votes
1answer
43 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, ...
-1
votes
1answer
84 views

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

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 $K \subset H$.