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

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15 views

Is membership in the index 2 subgroup of $Sp_4(\mathbb{F}_2)$ detected by a polynomial in the matrix entries?

I learned from Magma that $Sp_4(\mathbb{F}_2)$ has an index-2 subgroup isomorphic to $A_6$. Is it possible, given a matrix $M\in Sp_4(\mathbb{F}_2)$, to detect membership in this subgroup using a ...
1
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4answers
31 views

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$ To prove an equivalence relation my guess is to show that reflexivity, ...
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0answers
10 views

Coset multiplication is not well defined in the case of $S_4$

I have to show that coset multiplication is not well defined in this case. I have to choose 2 cosets $aH$ and $bH$ and locate two different representatives in each coset $a, a' \in aH$ and $b,b' \in ...
0
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0answers
35 views

if each $(i,j)\space:\space g_ig_j=g_jg_i$, then $G$ is abelian [on hold]

Let $G$ be finite group. say that $a,b\in G$ hold that $(a,b)\in R\subseteq G\times G$ iff $\exists g\in G \space:\space gag^{-1}=b$ note that $R$ is an equivalence relation. let $g_1,...,g_n$ be the ...
1
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0answers
11 views

Question about isotypical components

Consider $V=\bigotimes^3(\mathbb{C}^2)$ as a $\mathfrak{S}_3$ representation. One of its isotypical component is $S^3(\mathbb{C}^2)$, which is a linear subspace of symmetric tensors of ...
1
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2answers
28 views

Let $p,q$ are distinct primes and $G$ be a group of order $pq$ then which of the following is true?

Let $p,q$ are distinct primes and $G$ be a group of order $pq$ then which of the following is true? $1.G$ has exactly $4$ subgroups upto isomorphism. $2.G$ is abelian. $3.G$ is isomorphiq to a ...
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2answers
33 views

If $\ker(f) \neq \{0\}$, then $f(G)$ is abelian

Suppose $G$ is a group of order $15$ and $H$ is any group. Show that if $f:G \to H$ is a homomorphism with a non-trivial kernel (i.e. $\ker(f) \neq \{0\}$), then $f(G)$ is abelian). If $f$ is a ...
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1answer
30 views

Without using the fundamental theorem of homomorphisms show $G/(\ker(f)) \equiv H$

Let $G=\mathbb{Z}_9$, $H=\mathbb{Z}_3$, and $f(n)=n \mod 3$. Show $f:G \to H$ is well defined. Show $f:G \to H$ is an onto homomorphism. Without using Theorem 8.13 show $G/(\ker(f)) \equiv H$. ...
3
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3answers
60 views

Suppose that $H$ is a subgroup of $G$ and that $h$ and $h'$ are in $H$. If $h$ and $h'$ are conjugates in $G$, are they also conjugates in $H$?

Suppose that $H$ is a subgroup of $G$ and that $h$ and $h'$ are in $H$. If $h$ and $h'$ are conjugates in $G$, are they also conjugates in $H$? Why or why not? I have proved that the converse is true. ...
1
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1answer
22 views

Minimal normal subgroup that is not simple

Let $G$ be a nontrivial finite group. Then $H$ the intersection of all nontrivial normal subgroups has the property that if $K$ is a normal subgroup of $G$ such that $K \leq H$, then $K = H$ or $K$ is ...
1
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1answer
33 views

Can we define a binary operation on $\mathbb Z$ to make it a vector space over $\mathbb Q$?

It is known that any infinite cyclic group , in particular $(\mathbb Z, +)$ , can never be a vector space . So we may ask , Can we define an operation $*$ on $\mathbb Z$ such that $(\mathbb Z , *)$ ...
2
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1answer
19 views

Let $S$ be a subset of $R$ such that we have associative relation $*$ defined on $S\times S\to S$ with some properties…

Let $S$ be a subset of $\\R$ such that we have associative relation $\\*\\$ defined on $S\times S\to S$ with $$a*b*a=b \hspace{0.5cm} \forall a,b\in S, \hspace{1.5cm} \exists e \hspace{0.5cm} ...
3
votes
0answers
16 views

Solvable subgroup of group Aut(Zp+Zp)

Is it true that for any Solvable subgroup $G$ of group $Aut(\mathbb{Z}_p\oplus\mathbb{Z}_p), G^{(4)} = \{e\}$? Is it true for subgroups of $Aut(\mathbb{Z}_{p^n}\oplus\mathbb{Z}_{p^n})$? My idea is to ...
2
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1answer
33 views

is this set closed under addition?

I have some revision questions in my maths books and I'm a bit stuck on this one. Is $S=\{n^2:n \in \mathbb{Z}\}$ closed under the usual addition. I know that for it to be closed the sum of any 2 ...
7
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0answers
64 views

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

Where can I find a description and proof of the basic structure of $\text{PGL}_2(\mathbb{F}_p)$ (number of elements with each order, conjugacy classes, etc.) which is understandable by an ...
0
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1answer
24 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 a simple proof.
0
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1answer
22 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.
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2answers
25 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 ...
4
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3answers
72 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 ...
4
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1answer
61 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
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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
41 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 $\operatorname{SL_2}(\mathbb Z)$ ...
0
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1answer
43 views

Let $|G|=17.\;$ How many non-isomorphic subgroups of G are there? [on hold]

I don't know how to find non-isomorphic subgroups of a group. Thanks.
3
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1answer
44 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
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2answers
42 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 ...
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0answers
55 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
18 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
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2answers
248 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 ...
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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 ?
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3answers
39 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} ...
2
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1answer
24 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
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0answers
33 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 ...
7
votes
1answer
50 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
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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
32 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
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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 ...
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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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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 ...
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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 ...
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3answers
87 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
49 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
30 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 ...
1
vote
1answer
36 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
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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
321 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
votes
2answers
25 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 ...