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

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7 views

What are the irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$?

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
2
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1answer
31 views

Order of any element divides the largest order.

Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect ...
2
votes
1answer
13 views

$V^{\oplus3}$, linear constraints.

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$, and let $W = V \oplus V \oplus V$. Prove that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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2answers
23 views

all the squares in the multiplicative group $\mathbb{Z}_n^*$. [on hold]

I just want to know what this statement means: "all the squares in the multiplicative group $\mathbb{Z}_n^*$."
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0answers
26 views

$V$ is $G$-irrep. over $\mathbb{C}$, submodules of $V \oplus V \oplus V$ given by imposing linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$. Let $W = V \oplus V \oplus V$. Show that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
0
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1answer
18 views

I need example to satisfy this lemma: Let $P$ be a $p$-group and let $N$ be a nontrivial [on hold]

I need example to satisfy this lemma: Let $P$ be a $p$-group and let $N$ be a nontrivial, elementary abelian normal subgroup of $P$ which has a complement $X$ in $P$. If $P = \langle y \rangle X$ for ...
3
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1answer
37 views

Group of order $pqr$ and cyclic subgroup

Let $G$ be group of order $pqr$, when $p,q,r$ are different prime numbers. Does $G$ must have normal cyclic subgroup $H$ such that $G/H$ is cyclic too ? I know that $G$ has normal sylow subgroup of ...
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1answer
19 views

Mordell's theorem-Finitely generated abelian group

In my lecture notes we have the following: Mordell proved the following: Let $C$ be a nonsingular cubic curve with rational coefficients. Then the abelian group of rational points on $C$ is ...
2
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1answer
22 views

centre of a group presentation

having trouble showing that an element belongs to a centre of a group presentation. Let $G = \langle x,y,z\mid x^2=y^3=z^3=xyz\rangle$ I have to show that $ a = xyz$ belongs to the centre of $G$. I ...
3
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1answer
25 views

Let $H$ be a subgroup of the group $(R, +)$ such that $H$ $∩$ [-1,1] is a finite set containing a non zero element. Show that $H$ is cyclic.

Observations: Since $H$ is a subgroup of $(R, +)$ so $0 \in H.$ If $1 \in H,$ then all positive integers belong to $H.$ But $H$ is closed wrt addition, so the negative integers must belong to $H$ ...
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1answer
25 views

permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$,

Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. ...
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3answers
39 views

Building a proper homomorphism between groups.

Suppose I have a cyclic group $G$ of order $6$. I want to show that it is isomorphic to $\Bbb {Z}_6$. So $G=\{e,g^2,g^3,g^4,g^5\}=\langle g\rangle$. Can I build a homomorphism $f:G \to \Bbb{Z}_6$ that ...
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0answers
24 views

Order of $\frac{2}{3}$+Z in Q/Z

Let Q/Z be quotient group of addive group of rational numbers. Find order of element $\frac{2}{3}$+Z in Q/Z. I tried it by using facts that any G/H of G has induced operation from G. So I can do ...
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2answers
40 views

If $ H , K$ are subgroups of $G$ and $HK$ is subgroup of $G$ then $|H|$, $|K|$ aren't co-prime?

The statement I'm trying to understand is as written in the title : If $ H , K$ are subgroups of $G$ and $HK$ is subgroup of $G$ then $|H|$, $|K|$ aren't co-prime? I tried to find a counter ...
2
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1answer
22 views

Looking for example of a surjective homomorphism on $(\mathbb R,+)$ which is not an automorphism

Give example of a surjective function $f:\mathbb R \to \mathbb R$ such that $f(x+y)=f(x)+f(y) , \forall x,y \in \mathbb R$ but $f$ is not injective . I think I have to do something with basis of ...
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2answers
17 views

Prove that there is a fixed point in any subgroup $H$ of $S_4$ of order $6$.

Prove that in every subgroup $H$ of $S_4$ of order 6 there is a fixed point in {$1,2,3,4$}, i.e, there exists $1\le i\le 4$ such that $h(i)=i$ $\forall h\in H$. $Start$: Suppose there is a subgroup ...
2
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1answer
32 views

Does pigeonhole principle apply for all groups?

I'm reading Rosen's book and it has a proof to show that a finite subgroup (set) is closed under a composition law. It says for some $i$ and $j$, $i < j$, $a^i = a^j$ i.e, $a^i = a^i \circ ...
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1answer
21 views

class equation of order $10$

Is it a class equation of order $10$ $10=1+1+1+2+5$. As far as I know for being a class equation each member on RHS has to divide $10$ and should have at least one $1$ on RHS, which is ...
2
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0answers
22 views

A question about group actions on a trees

why does the following conclusion hold: Let G be a group acting on a tree $\Gamma$, H a subgroup of G with minimal subtree $\Gamma_H$ and $g\in G$ be a hyperbolic element, s.t. $\langle g\rangle\cap ...
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4answers
33 views

Number of onto and into group homomorphisms between $\mathbb Z$ and $\mathbb Z$

How many homomorphisms are there of $\mathbb Z$ onto $\mathbb Z$ $\mathbb Z$ into $\mathbb Z$ These two questions are from exercise 13, from book by John B. Fraleigh. Answer of 1. is "two ...
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15 views

Number of homomorphisms between G TO/INTO/ONTO G'

How many homomorphisms are between following groups: Type 1. $Z_{10}$ to $Z_{10}$ Type 2. $ Z$ to $Z_{10}$ Type 3. $ Z$ onto $Z $ Type 4. $ Z $ into $ Z $ Type 5. $Z$ into $Z_2 $ These type of ...
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1answer
35 views

For a group G define the set $Z(G)$ by [on hold]

$Z(G) = \{ fz \in G\mid zg = gz \;\forall g \in G\}$ . In other words, Z(G) is the set of all elements that commute with every other element. Show that Z(G) is an abelian subgroup of G I understand ...
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1answer
32 views

If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$.

If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$. I cannot get heads nor tails of how to even start this let alone finish ...
2
votes
0answers
23 views

Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
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3answers
47 views

The elements of $\Bbb{Z}_{20}^{\times}$

The elements of $\Bbb{Z}_{20}^{\times}$, as I understand, are all the number from 1 to 20 included that are relatively prime to 20? I am having troubles finding a coherent definition of this kind of ...
3
votes
1answer
68 views

Finding $10\otimes 8\otimes 8\otimes 8$ in $SU(3)$

I know that in $SU(3)$ $$8\otimes 8 = 27+10+\bar{10}+8+8+1. $$ How can one use this to compute $$10\otimes 8\otimes 8\otimes 8?$$ Can one start with simplifying $$\tag{1} 10\otimes 8\otimes 8 = ...
3
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2answers
38 views

Prove that for every $x$ in a group $G$ there is a $y$ such that $y^n=x$.

Let $G$ be a finite group and let $n$ be a natural number, relatively prime to $|G|$. Prove that for every $x$ in a group $G$ there is a $y$ such that $y^n=x$. I really need assistance when it comes ...
2
votes
3answers
96 views

On special normal subgroup of a group

Let $G$ be a group and $H$ be a subgroup of $G$ such that for any $x\in G$ we have $x^2\in H$. prove that $H$ is normal in $G$. I think this true, but can not prove it. for example this is true for ...
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1answer
54 views

Extension of Goursat's Lemma

Consider $n \geq 2$ groups $G_1,..., G_n$ each having no non-trivial abelian quotient, and let $H$ be a subgroup of $G_1 \times ... \times G_n$ such that every projection map $H \to G_i \times ...
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1answer
39 views

Is the distribution of the orders of the cyclic groups generated by elements of $S_n$ known?

A week ago I was playing around with a card-shuffle method corresponding to an element of $S_{52}$, and the order of the cyclic group generated was 272 (ie, 272 shuffles returns the deck to original ...
2
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0answers
13 views

About a finite subgroup generated by a finite set of conjugates of a element

Let $G$ be a group. Let $H$ a subgroup of $G$ such that have no subgroups of finite index. Suppose that exist a element of finite order, say $a$ such that $G = \langle a, H \rangle$. Suppose that ...
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1answer
19 views

How two cosets are same in this example

I was reading about cosets from example 298 in this pdf. I repeat the problem here: Let $G = \{e, a, a^2, a^3\} = <a>$ where $|a| = 4$. Let $H = \{e,a^2\} = < a^2 >$ Then ...
3
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0answers
10 views

Uniqueness of induced functions on reduced free groups

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and let $K_n$ be the reduced free group, that is, $F_n$ modulo the relation that $[x_i,x_i^g]=1$ for all $i\in\{1,\cdots,n\}$,$g\in F_n$, ...
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0answers
30 views

Semigroups and solutions of equation

It is easy to prove: in a finite semigroup if for all $a$ and $b$, $ax=b$ and $ay=b$ has unique solution. then it is group. But if in a finite semigroup, if for all $a$ and $b$, $ax=b$ and $ay=b$ has ...
3
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1answer
41 views

Describe the subgroup $K\leq S_4$ of order 8

How do I construct the subgroup $K$ (a subgroup of $S_4$ of order $8$) ?
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1answer
18 views

one end group with positve first Betti number$\beta^{(2)}_1(G)>0$

Could anyone give me an example of a countable finitely generated (f.g.) discrete group $G$ with one end but have non-trivial $H^1(G,\ell^2G)$? To be precise, consider the following two cases. (1) ...
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0answers
37 views

Direct products of simple non-abelian groups 2

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ a simple non-abelian subgroup of $G$ such that $Z(G)\leq N$ and $\frac{G}{N}\cong A$ be a non-abelian simple group. Is it true that $G=N\times A$ ...
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28 views

Abelian minimal normal subgroup in a finite non-solvable group

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ an abelian minimal normal subgroup of $G$ ( $|N|=p^d$ for some integer $d$ and prime $p\neq 2,3,5$) such that $N=C_G(N)$, $Z(G)\leq N$ and ...
2
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2answers
48 views

Subgroup of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ where $(m,n)=1$.

Let $m,n>1$, $(m,n)=1$. Prove that every subgroup $H$ of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ is $H=A\oplus B$ where $A=H\cap \Bbb {Z}_n$ and $B=H\cap \Bbb {Z}_m$. First attempt: $G=\Bbb {Z}_m \oplus ...
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1answer
29 views

cohomology of semi-direct product of groups

Let $G, H$ be groups. Let $G\rtimes _\phi H$ be a semidirect product. The product is twisted. Let $BG$, $BH$, and $B(G\rtimes_\phi H)$ be the classifying spaces of $G$, $H$, and $G\rtimes _\phi H$. ...
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0answers
13 views

What is the Frobenius element (if any) of this group.

For $F = \mathbb{Q}[\sqrt{5}]$ with $p = 2$ and prime ideal over p of $ q = (2, 1 + \sqrt{5})$ with the Frobenius element defined as $$ x^{Frob_q} \equiv x^p (mod q) $$ with $Frob_q \in Gal ...
2
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0answers
32 views

Some subgroup of $GL_2(\mathbb{Q})$

Let's consider $GL_2(\mathbb{Q})$ and $C_2\times C_2 \times C_2$, $C_2$ - cyclic group of order 2. I can't show, that group $C_2\times C_2 \times C_2$ is not a subgroup of $GL_2(\mathbb{Q})$.I don't ...
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1answer
28 views

Cardinality of HK

Let $G$ be a group and let $H$ and $K$ be two subgroups of $G.$ If both $H$ and $K$ have $12$ elements which of following numbers cannot be cardinality of set $HK=\{hk:h \in H, k \in K\}$? 1.72 2.60 ...
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1answer
35 views

Show that $H$ is normal subgroup of $G$.

Let $H\leq G$. Show that $H$ is normal iff $xHx^{-1}=H\space \forall x\in G$. My textbook defines normal subgroup of $G$ as kernel of some homomorphism which has $G$ as domain. I showed that if $H$ ...
2
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1answer
39 views

Why is every coset in G a subset of G?

Suppose $G$ is a group and $H$ is a subgroup of $G$. $Ha$ is a right coset of $H$ in $G$. According to the Dover Book of Abstract Algebra p. 127, "Every coset in $G$ is a subset of $G$." I understand ...
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2answers
26 views

Generators of $ D_8$

Let G= $ D_8$ be dihedral group of symmetries of square. Find the minimal number of generators for G. My book directly writes thar answer is 2. In order to do this do we have to remember the group ...
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3answers
38 views

group homomorphisms from the real line to infinite torsion abelian groups

Hello I have question in group theory that actually originated from a question in dynamical systems. Let G be the abelian group given by the real line with addition. Let H be an infinite torsion ...
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2answers
30 views

cyclic groups -class is prime number

How can I prove that a group $G$, such that $|G| = p$, where $p \in \mathbb{P}$, is cyclic?
3
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0answers
35 views

Showing $G/Z(R(G))$ isomorphic to $Aut(R(G))$

I am working on this problem with lots of nesting definitions: Show that $G/Z(R(G))$ is isomorphic to a subgroup of $Aut(R(G))$. For your info, $R(G)$ is called the Radical of $G$, defined as ...
2
votes
1answer
39 views

In a group of order 21, every normal subgroup is cyclic [on hold]

Let $P$ be a group of order $21$. How to prove that each normal subgroup of $P$ is cyclic?