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

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-3
votes
0answers
33 views

Bigenetic properties of finite group [closed]

Nilpotency, supersolubility and polycyclicity are bigenetic properties of the class of all finite group. Let be : P is property, X be a class o group. We say that P is a bigenetic property of ...
-2
votes
0answers
27 views

orbits from group [closed]

Let g is the group Z2 and let 56 points as follows: w:= [ 7, 8, 15, 27, 42, 89, 95, 121, 125, 134, 139, 150, 167, 5 , 10, 11, 18, 30, 45, 92, 98, 124, 128, 137, 142, 153, 170, 8 , 12, 13, 20, ...
1
vote
1answer
53 views

Number of units of $\mathbb{Z}/11\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$

Let $n\mathbb{Z} = \{nk\::\:k \in \mathbb{Z} \}$. Find the number of units of $\mathbb{Z}/11\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$. I tried this problem by using the fact that since ...
1
vote
0answers
25 views

What is the root structure of the Diffeomorphism Group?

Being a physicist, I think it'd be cool to have Coxeter plane projections of the root systems of the symmetry groups associated with the fundamental forces hanging on my walls (example for E8: ...
0
votes
1answer
43 views

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

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
3
votes
2answers
81 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
31 views

$V^{\oplus3}$, linear constraints. [closed]

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 ...
0
votes
2answers
30 views

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

I just want to know what this statement means: "all the squares in the multiplicative group $\mathbb{Z}_n^*$."
-2
votes
0answers
33 views

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

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
votes
1answer
30 views

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

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 ...
4
votes
1answer
53 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 ...
1
vote
1answer
26 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
votes
1answer
65 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 ...
4
votes
2answers
66 views
+50

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 $H$ ...
1
vote
1answer
27 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. ...
1
vote
3answers
42 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 ...
0
votes
0answers
30 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 ...
0
votes
2answers
51 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
votes
1answer
27 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 ...
1
vote
3answers
33 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
votes
1answer
34 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 ...
1
vote
1answer
32 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
votes
0answers
30 views

A question about group actions on a trees [on hold]

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 ...
1
vote
4answers
40 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 ...
0
votes
0answers
16 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 ...
-2
votes
1answer
36 views

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

$Z(G) = \{ fz \in G\mid zg = gz \;\forall g \in G\}$ . 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 ...
0
votes
1answer
36 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 ...
3
votes
1answer
39 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. ...
1
vote
3answers
55 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 ...
4
votes
1answer
115 views

Finding $\mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8}\otimes \mathbf{8}$ in $SU(3)$

I know that in $SU(3)$ $$\mathbf{8}\otimes \mathbf{8} = \mathbf{27}+\mathbf{10}+\mathbf{\bar{10}}+\mathbf{8}+\mathbf{8}+\mathbf{1}. $$ How can one use this to compute $$\mathbf{10}\otimes ...
3
votes
2answers
46 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
106 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 ...
-2
votes
1answer
61 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 ...
5
votes
1answer
43 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
votes
0answers
16 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 ...
0
votes
1answer
22 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
votes
0answers
11 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 $i\in\{1,\cdots,n\}$,$g\in F_n$, ...
5
votes
0answers
52 views

Semigroups and solutions of equation

It is easy to prove: in a finite semigroup if for all $a$ and $b$, $ax=b$ and $ya=b$ has unique solution. then it is group. But if in a finite semigroup, if for all $a$ and $b$, $ax=b$ and $ya=b$ has ...
3
votes
1answer
51 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 $8$) ?
0
votes
1answer
19 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) ...
1
vote
0answers
41 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 $G=N\times A$ ...
1
vote
0answers
32 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
votes
2answers
52 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 ...
1
vote
1answer
32 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 $G\rtimes _\phi H$. ...
0
votes
0answers
14 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
votes
0answers
36 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 ...
0
votes
1answer
34 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 ...
1
vote
1answer
39 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
votes
1answer
46 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 ...
-1
votes
2answers
32 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 ...