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

learn more… | top users | synonyms (2)

0
votes
3answers
27 views

Question about group theory and order of elements

Let $G$ be a group and $x, y \in G$. Prove that $ord(x)=ord(y^{-1}xy).$ Let $n,m$ be integers such as $x^n=1$ and $(y^{-1}xy)^m=1$. $x^n=(y^{-1}xy)^m=y^{-1}x^my=1$ I'm not sure how should I ...
0
votes
0answers
25 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
-2
votes
0answers
15 views

Bigenetic properties of finite group [on hold]

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
24 views

orbits from group [on hold]

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, ...
0
votes
1answer
45 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 ...
0
votes
0answers
18 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
21 views

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

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
2
votes
2answers
63 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
19 views

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

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
25 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 $\mathbb{Z}_n^*$."
-2
votes
0answers
29 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
votes
1answer
20 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
votes
1answer
41 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 ...
0
votes
1answer
21 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
25 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
votes
1answer
28 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 $H$ ...
1
vote
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. ...
1
vote
3answers
40 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
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 ...
0
votes
2answers
42 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
23 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
2answers
21 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
33 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 ...
0
votes
1answer
23 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
25 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 ...
1
vote
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 ...
0
votes
0answers
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 ...
-2
votes
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 ...
0
votes
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
24 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
48 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
71 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
votes
2answers
40 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
98 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
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 ...
5
votes
1answer
40 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
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 ...
0
votes
1answer
20 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
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 $i\in\{1,\cdots,n\}$,$g\in F_n$, ...
6
votes
0answers
35 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
votes
1answer
44 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
38 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
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
votes
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 ...
1
vote
1answer
31 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
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
votes
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 ...
0
votes
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 ...
1
vote
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 $H$ ...