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

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Basic question about dimensionality of Euclidean group

I have a basic question about the dimensionality of the Euclidean group. Why are degrees of freedom greater than the dimension? I thought that a degree of freedom is the same as a dimension, as in, ...
1
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1answer
29 views

If G is a finite group and $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$

Let G be a finite group. Show that, given $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$. I'm trying to use the info that is finite, but I can't find a way. For instance, if G is a ...
3
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2answers
27 views

What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes? I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, ...
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0answers
15 views

Groups - Compositions

If the f is written to the right of its argument does that mean the composition of $f g$ is actually $g(f(x))$ instead of being $f(g(x))$ which is the notation I'm used to. I ask this because I read ...
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31 views

An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
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1answer
19 views

Lower bound of the index of a subgroup of a non abelian simple group

Let $G$ be a simple , non abelian group . Let $H$ be a subgroup of $G$ such that $[G:H] < \infty$ . Show that: $[G:H] \ge 5$
3
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1answer
15 views

Derived subgroup of the base group of a standard wreath product

Let $G=H\wr K$ be the standard wreath product with $K\ne 1$. Prove that $B'\leq [B,K]$ where $B$ is the base group of $G$. Deduce that $G/[B,K]\cong (H/H')\times K$. This is problem 1.6.20 from ...
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0answers
23 views

What is the difference between operator groups and group actions?

I have always thought that operator groups and group actions are two names for the same thing. Now I have noticed that they have different codomains. Actually I am still confused, why are they ...
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4answers
27 views

Groups - Inversions

Above is just an example I'm trying to work from as I have the solutions. I've seen lots of definitions of what inversions are but they use signs like sigma, and it doesn't really explain what the ...
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0answers
26 views

$|G'/G''| \geq p^3$ where $G$ is $p$ group [on hold]

Let $G$ be a $p$ group. Commutator subgroup of $G$ is denoted by $G'$ prove that 1. If $G'$ has order $p^3$ ,then $G'$ is abelian 2. If $G''$ is not identity then $|G'/G''| \geq p^3$
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35 views

Can these two quotient groups be isomorphic?

Let $N$ and $M$ be two normal subgroups of a group $G$. Then we can show that the set $NM \colon= \{\ nm \ | \ n \in N, \ m \in M \ \}$ is a subgroup of $G$, that $M$ is normal in $NM$, and that $N ...
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1answer
32 views

conjugacy classes and order of group

Suppost that $k_G(A)$ denotes the number of conjugacy classes of $G$ that intersects $A$ non-trivially ($A$ is an arbitrary subset of $G$) and $M=G^{'}Z(G)$. Also suppose that $G$ is non-solvable, ...
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2answers
33 views

Is every normal subgroup the kernel of some endomorphism?

Let $G$ be a group, and let $N$ be a normal subgroup of $G$. Then there is the canonical homomorphism $\phi$ of $G$ onto $G/N$ with kernel $N$. This homomorphism is defined as follows: $\phi(g) ...
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0answers
16 views

Conjugacy classes of right cosets

Is it true that all elements of a right coset $Hx$, for a subgroup $H$ of $G$, contained in a unique $G$-conjugacy class? I mean if $Hx=\lbrace{x_1,...,x_s}\rbrace$, then is it true that ...
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1answer
23 views

permutation problem: cycle representation

Let $n$ be an odd number. Let $C_n$ be the set of permutations $\pi$ of $[n]$ whose cycle representation has only one cycle. Let $\pi,\sigma\in C_n$. Prove that their composition $\pi\sigma$ has an ...
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0answers
10 views

Questions on proof that transvections are conjugate in $GL(V)$.

I have difficulty following the proof that transvections are conjugates in $GL(V)$, and for $n \ge 3$ even in $SL(V)$. I give the necessary definitions and the proof, with the problematic parts ...
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1answer
61 views

If $xy=x^{-1}y^{-1}$, does this imply $x=x^{-1}$

This seems like a simple enough question, trying to show that if the title condition holds, that a group $G$ of which $x,y$ are elements, then $G$ is Abelian. $$xy=x^{-1}y^{-1}=(yx)^{-1}$$ From here I ...
4
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1answer
65 views

If $[G' : G'']\leq p^2$, then $G'$ is abelian.

Problem : Let $G$ be a p-group and $G'$ denote the commutator subgroup of $G$. If $[G' : G'']\leq p^2$, then $G'$ is abelian. It is easy to prove it for the case of $[G' : G'']=1$ since G is ...
4
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2answers
68 views

Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group ...
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22 views

Groups with single conjugacy class of subgroups.

I wish to know all those groups in which there is single conjugacy class of subgroups of fixed order.For example, In finite cyclic groups and in Alternating group of degree 4, number of conjugacy ...
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1answer
22 views

Find all sub groups of order $4$ in $Z_4 \bigoplus Z_4$ . Are they all cyclic?

Find all sub groups of order $4$ in $Z_4 \bigoplus Z_4$ . Solution : $Z_4 =\{0,1,2,3\}$ $O(1) = O(3) = 4 , O(0) = 1 ; O(2) = 2$ Hence, i found the subgroups of order $4$ as follows : $\langle 1,0 ...
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2answers
90 views

A group with finitely many subgroups must be a finite group

Show that a group that has only a finite number of subgroups must be a finite group.(Fraleigh, A First Course in abstract Algebra-7th Edition,pg.67) I could not show properly so I need help. Thank ...
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0answers
25 views

what is the conjugate of irreducible character of $G\wr S_n $?

Assume $G$ is a finite group and field as a complex field. As we know that, the index set of representations of $G\wr S_n$ is set of all $k$-tuble of partitions ...
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2answers
23 views

Cartesian Product of Sets and the Direct Product of Groups

I'm having a bit of confusion. I've tried to search youtube and whatnot but I could not find any explanations. My book says the following: The Cartesian Product is denoted by: $$S_1\times ...
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2answers
41 views

Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
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0answers
11 views

Roots and Weights

I use a Mathematica package to compute roots and weights (and other things) but the package gives me only the expression of the roots in $\omega$-basis (basis of fundamental weights) and in the ...
0
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1answer
29 views

Groups/Sets Notation Question

Simple question: But what does the sigma small Y mean, does it just represent a group? Also have seen this with numbers, and not quite sure what it means. Thanks
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22 views

Schur-Weyl duality from Double Commutant Theory

Let $V$ be a finite dim complex vector space. Then $V^{\otimes n}$ carries an action by $S_n$ by permuting factors $\sigma(\pi)(v_1\otimes...\otimes v_n)=v_{\pi^{-1}(1)}\otimes...\otimes ...
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3answers
32 views

If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.

I'm struggling with this. I know it is going to use Lagrange's Theorem so this is what I have so far: Suppose $|U_n| = k$ This implies $a^k = 1$ for all a in $U_n$ and $|a|$ divides $k$. Now, what ...
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1answer
36 views

Finding Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$. The mistake in this method?

We need to find the Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$ . This method does not give me the right answer (i.e $6$ ) . Attempt: We need to find the number of Cyclic ...
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1answer
24 views

Automorphisms of group extensions

Assume we have a group extension $1 \to N \to G \to H \to 1$, and an automorphism $\phi: G \to G$. Is it correct that this automorphism induces automorphisms $\phi_N : N \to N$ and $\phi_H : H \to H$ ...
3
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1answer
58 views

A question about groups generated by two elements.

Suppose a group $G=\langle a,b \rangle$ and $|G|<\infty$ where $|a|=m_0$ and $|b| = m$. How is it that the operation table for $G$ can be completely determined just by knowing $ab=b^na$ for some ...
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4answers
94 views

Do we have $(G/H)\times H \cong G$ for groups in general?

After some thought I began to suspect $(G/H)\times H \cong G$, so I tried to construct an isomorphism by hand. I came up with $\varphi: (gH, h) \mapsto gh$ which came out to work provided $G$ is ...
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1answer
20 views

the smallest quasigroup, which is not a group

I'm wondering, which is the smallest quasigroup, which is not a group? And how to check it?
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1answer
50 views

Isomorphisms between finite abelian groups and cyclic groups

If G is abelian of order 175 and H is cyclic of order 25 and there is a homomorphism from G onto H then what is G isomorphic to? I can see how G is isomorphic to either $C_{25} * C_7$ or to $C_5 * ...
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1answer
59 views

Isomorphisms in finite abelian groups

Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to? I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and ...
0
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1answer
27 views

Range and kernel of groups

Let $f: G \rightarrow H$ be a homomorphism. If the range of $f$ has $n$ elements, then $x^n \in$ ker $f$ for every $x \in G$. I can kind of understand why this is true. The ker of $f$ is $\{x \in ...
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1answer
37 views

Cyclic and abelian group

A group $G$ has order $25\cdot 47\cdot 17$. Is it cyclic and/or abelian? I know that a group of order $47$ or $17$ is cyclic, should I somehow use it?
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1answer
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Normal subgroup $N$, subgroup $U$, then $UN/N = U/N$.

Let $G$ be a group and $N \unlhd G$ a normal subgroup, $U \le G$ some subgroup. Then I guess $U / N$ is always some group, and moreover $U / N = UN / N$, because $UN / N = \{ unN : u \in U, n \in N \} ...
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2answers
47 views

Non-abelian group of order 28 which is not the dihedral group

Consider the group of order 28 with Sylow $2$-subgroups that are cyclic. We can derive that the Sylow $7$-subgroup is normal, and that this group is uniquely determined by the relation $bab^{-1}=a^6$ ...
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1answer
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If $G$ is solvable and $G/[G,G]$ is cyclic, can $G\times G$ be generated by 2 elements?

I doubt this is true, but I haven't found any small counterexamples (there are no counterexamples with $|G| < 1536, |G| \neq 768$): Suppose $G$ is finite, solvable, 2-generated, and $G/[G,G]$ ...
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1answer
26 views

A group generated by two elements such that its product with itself is not generated by two elements.

We have $S_5=\langle (12345), (12)\rangle$ and we can show that $S_5\times S_5$ is also generated by two elements. Is there a group $G$ generated by two elements such that $G\times G$ is not generated ...
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0answers
35 views

Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
4
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0answers
25 views

Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
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0answers
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Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. Condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle ?$

Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. State a necessary and sufficient condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle$ Attempt : Let $l = ...
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1answer
31 views

$[G:\cap H_i]\leq\Pi[H_i:H_{i+1}]$

If $H_0=G$ and $H_{n+1}\subseteq H_n\subseteq G$ for $n\in \mathbb N$, then $[G:\cap H_i]\leq\Pi[H_i:H_{i+1}]$. I used Poincare inequality, but it doesn't work.
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2answers
45 views

Inn characteristic in Aut

If $G$ is a centerless group then is $\mathrm{Inn}(G)$ necessarily characteristic in $\mathrm{Aut}(G)$? The condition of being centerless is necessary as $D_8$ provides a counterexample otherwise.
2
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1answer
62 views

Introducing multiplication of cosets

So, i have encountered two ways to introduce the multiplication of cosets, and i want to understand exactly what is happening in each, specifically in light of the multiplication of cosets being ...
3
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20 views

Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
3
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2answers
67 views

What is the reason for stating Cayley's theorem this way?

In my notes, Cayley's theorem reads: Any group $G$ is isomorphic to a subgroup of $\text{Sym}\, X$ for some $X$. On the other hand, several sources (such as Wikipedia) give a slightly more ...