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

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Construction of representations

Is there an example, where given a conjugacy class in a finite group, can we construct an irreducible representation from it?
3
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1answer
144 views

Conjugacy classes of maximal tori in PGL2: only 2?

If $K$ is a field, and $F$, $L$ are two distinct quadratic extensions of $K$, then must the subgroups of $\operatorname{PGL}(2, K)$ defined by $F$ and $L$ be conjugate? To define the subgroup, ...
2
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1answer
499 views

Isomorphism on commutative diagrams of abelian groups

Consider the following commutative diagram of homomorphisms of abelian groups $$\begin{array} 00&\stackrel{f_1}{\longrightarrow}&A& \stackrel{f_2}{\longrightarrow}&B& ...
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2answers
269 views

commutative diagrams of group homomorphisms

consider the commutative diagram of group homomorphisms: $$\begin{matrix} A&\stackrel{f}{\rightarrow}&B\\ \downarrow{g}&&\downarrow{k}\\ C&\stackrel{h}{\rightarrow}&D ...
1
vote
1answer
124 views

writing a field as an R module

let $F$ be a field. for which ring $R$, $F$ is an $R$-module. i know already that as an abelian group $F$ is a $\mathbb Z$- module, what else can we say for a general field $F$.
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1answer
90 views

Product of simple groups has no proper characteristic subgroups?

Let S be a simple group and set $G = S \times ... \times S$. I believe that G has no proper characteristic subgroups but I have no idea about how to prove this. Any help or counterexamples would be ...
3
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1answer
97 views

Which are the infinite balanced nonabelian groups?

This question is motivated by this previous question. A group $G$ is called a balanced group provided that for all $a,b \in G$, either $ab=ba$ or $a^2 = b^2$. Following the answers provided by ...
2
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2answers
269 views

Example of an infinite balanced nonabelian group

A group is called a balanced group provided that, for all $a,b \in G$, either $ab=ba$ or $a^2=b^2$. Example of (possibly infinite) balanced groups are abelian groups. An example of a finite ...
7
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1answer
170 views

Sylow $p$-subgroups of $\operatorname{PGL}(2,K)$ for $K$ infinite and $p$ odd

Is every maximal $p$-subgroup of $\operatorname{PGL}(2,K)$ conjugate, where $p$ is an odd prime not equal to the characteristic of $K$? Here $\operatorname{PGL}(2,K)$ is the quotient group of the ...
4
votes
3answers
470 views

is every subgroup of a semi-direct product of groups a semi-direct product of subgroups?

the title pretty much says it all... anyway... let G be a semi-direct product of N by Q, and let H be a subgroup of G can one always find subgroups N1 and Q1, of N and Q respectively, such that H is ...
2
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1answer
129 views

$2×2=3+1$ for $\operatorname{GL}_2$

If $V$ is the natural representation for $\operatorname{GL}(2,q)$, then $V⊗V$ appears to decompose into the direct sum of a (strange?) one-dimensional module and a three dimensional module. I've ...
14
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2answers
567 views

finite subgroups of PGL(3,C)

The enumeration of finite subgroups of $\operatorname{PGL}_2(\mathbb{C})$ is one of the classic classification problems: mathematicians in many fields know well that the answer is cyclic groups, ...
3
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1answer
416 views

What is the difference between a “change of basis” and a “similarity transformation”?

In crystallography we define a "misorientation", $M_{AB/A}$, as the rotation required to bring crystal A into coincidence with crystal B, as measured with respect to the reference frame of crystal A. ...
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8answers
533 views

Order of cyclic groups

Wikipedia says: It is known that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if n is 1 or 2 or 4 or $p^k$ or $2p^k$ for an odd prime number p and k ≥ 1. The statement seems provable ...
2
votes
2answers
173 views

$X\!\supseteq\!K\!\simeq\!0\Rightarrow X\!\simeq\!X/K$ ($\pi_1$ of a connected graph is free)

How can I prove the following: If $X\supseteq K$ is contractible, then the quotient $X/K$ is homotopy equivalent to $X$? Since $K$ is contractible, we have a homotopy $H:id_K\!\simeq\!c_{k_0}$ ...
2
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1answer
137 views

A quotient group and its torsion elements

Let $(x_1,\dots,x_r)$ be a non-zero element of $\mathbb{Z}^r$, and let $h$ be the highest common factor of $x_1, \dots, x_r$. Show that: $$ \mathbb{Z}^r/\langle(x_1,\dots,x_r)\rangle \cong ...
3
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2answers
160 views

Every polycyclic group has a normal poly-infinite cyclic subgroup of finite index

There is a theorem that states: "Every polycyclic group has a normal poly-infinite cyclic subgroup of finite index. " I just read the proof of it and honestly found some difficulties in it. The main ...
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1answer
80 views

A needed counterexample in supersolvability

I need a counterexample for this fact that: If $G$ is supersolvable, so any quotient group of it, is not neccesarily supersolvable. A infinite one is prefered. Thanks
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votes
1answer
468 views

Sylow subgroups

Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $G=HK$. Show that there exists a $p$-Sylow subgroup $P$ of $G$ such that $P = (P \cap H)(P \cap K)$. It is not hard to prove that ...
1
vote
1answer
79 views

Is there a finite generating set for the Torelli group $T_2$?

D.Johnson showed in 1983 that for g>2 , the Torelli group $Tg$ has a finite set of generators. I have not been able to find out what the case is for g=1,2; does anyone know of any result for ...
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1answer
677 views

Augmentation ideal of the group ring

Let $G$ be a group and $I_G$ be the augmentation ideal of the group ring $\mathbb{Z}G$, i.e. $I_G$ consists of formal linear combinations $\sum n_i g_i$ ($n_i\in\mathbb{Z}$, $g_i\in G$) such that ...
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2answers
730 views

Completing an exact sequence

This seemingly simple question has puzzled me for a while: Determine which abelian groups A fit into a short exact sequence $$0 \to \mathbb{Z}_{p^m} \to A \to \mathbb{Z}_{p^n} \to 0$$ (This is ...
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4answers
408 views

If the size of 2 subgroups of G are coprime then why is their intersection is trivial?

Let H and K be subgroups of G, with size p and q respectively, where p and q are coprime, how can we show that H intersect K is {e} where e is the identity element in G
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1answer
68 views

factorisation of a trivial homomorphism

let $G$ be an abelian group. and $f:G\rightarrow \{0\}$ be the trivial homomorphism. suppose there exists $G\stackrel{g}{\rightarrow} H \stackrel{h}{\rightarrow} \{0\}$ such that $f=h\circ g$ does ...
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1answer
649 views

Group of even order contains an element of order 2

I am working on the following problem from group theory: If $G$ is a group of order $2n$ (i.e., the cardinality of the set $G$ is $2n$), show that the number of elements of $G$ of order $2$ is odd. ...
4
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1answer
93 views

A lemma about free groups

Let F be a finitely generated free group and $\gamma_m$ the lower central series. Why is $\gamma_m(F)/\gamma_{m+1}(F)$ torsionfree? I know it is abelian, but I couldn't find out more about it, as ...
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3answers
115 views

A type of isomorphism on $\mathbb{Z}^r$

Let $(x_1,\dots,x_r)$ be a non-zero element of $\mathbb{Z}^r$, and let $h$ be the highest common factor of $x_1, \dots, x_r$. Show that there is an isomorphism $\mathbb{Z}^r \to \mathbb{Z}^r$ taking ...
0
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1answer
142 views

What is this set of block matrices called?

Let $S$ be the set of matrices defined inductively as follows: The scalar $1 \in S$. $M \in S$ if $M$ is a $2 \times 2$ block diagonal matrix where the blocks are recursively in $S$. $M \in S$ if ...
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2answers
443 views

Example of a group

Can one give an example of a finite group $G$, with a subset $H$ containing identity, such that $gHg^{-1}=H$ for all $g\in G$, $|H|$ divides $|G|$, but $H$ is not a subgroup of $G$. Motivation ...
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4answers
1k views

A kind of converse of Lagrange's Theorem

Let $G$ a finite group. If $a\in G$, a consequence of the Lagrange's Theorem is that the order of $a$ divides the order of $G$. Let $p=|G|$. If $p$ is prime, it is well known that $G$ is cyclic, and ...
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votes
3answers
1k views

Combination problem with constraints

You have four containers and one pitcher of water that holds 100L. Each container has different capacities with maximums of, say...70L, 45L, 33L and 11L levels respectively. What is the formula that ...
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2answers
66 views

For $h \in G$ and $\phi \in Aut(G)$ is $\phi^n(h)$ periodic in any finite quotient, if $G$ is finitely generated?

This is new variant of For $h \in G$ and $\phi \in Aut(G)$ is $\phi^n(h)$ periodic in any finite quotient? thanks to Alon's comment. Since that is the case I'm interested really in, I figured it's ...
6
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1answer
60 views

For $h \in G$ and $\phi \in Aut(G)$ is $\phi^n(h)$ periodic in any finite quotient?

Let $G$ be an infinite group, and $\phi$ an automorphism of it. Let $N$ be a normal subgroup of $G$ such that $G/N$ is finite. Is it true that for any $h$ in $G$, $\phi^n(h)N$ (as a sequence of ...
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3answers
182 views

How do I construct this quasicyclic group?

Question: Let $p$ be a prime number. Let $G_n=C_{p^n}$ be the cyclic group of order $p^n$ with generator $x_n$. We define $\varphi:G_n \rightarrow G_{n+1}$ by $\varphi(x_n^a)=x_{n+1}^{pa}$. Using ...
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2answers
178 views

Constructing the semidirect product $C_{2^n}\rtimes C_2$.

Starting from the group of automorphisms of $C_{2^{n}}$ find the automorphism $x$ of order 2 (with $n\geq 3$) and building the semidirect product $C_2$ and $C_{2^{n}}$ by $y$, where $y$ is an ...
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2answers
581 views

Show that a finite group with certain automorphism is abelian

Let $G$ be a finite group and $f:G\to G$ an isomorphism. If $f$ has no fixed points (i.e., $f(x)=x$ implies $x=e$) and if $f\circ f$ is the identity, then $G$ is abelian. (Hint: Prove that every ...
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5answers
2k views

Examples and further results about the order of the product of two elements in a group

Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$? Wikipedia says "not much": There is no general formula relating the order of a ...
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2answers
109 views

$p$-th powers of elements of an extraspecial $p$-group

For $H$ a group and $n\in\mathbb{N}$, let $H^{(n)}=\langle h^n : h \in H \rangle$. Now let $G$ be an extraspecial $p$-group (see definition). Is it true that $G^{(p)}\cong \mathbb{Z_p}$. (It holds for ...
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1answer
214 views

When $G'$/$G''$ and $G''$ both are cyclic groups

There is a claim saying that if both $G'/G''$ and $G''$ are cyclic groups, then $G''=1$, where $G'$ is the derived subgroup of the group $G$. I have been thinking of ...
4
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2answers
263 views

is there a decomposition of $L_2(q)$ into a direct product?

I wonder if there is a nontrivial decomposition of the $L_2(q)$, where $q$ is a prime power, into a direct product. I think that there is none, but I am not sure. $L_2(q)$ refers to the special ...
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4answers
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Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian

I've been working on this problem listed in Herstein's Topics in Algebra (Chapter 2.3, problem 4): If $G$ is a group such that $(ab)^i = a^ib^i$ for three consecutive integers $i$ for all $a, b\in ...
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1answer
210 views

Bound for the rank of a nilpotent group

Let $G$ be a nilpotent group generated by $d$ elements. Is there a function $r(d)$ such that every (necessarily finitely generated) subgroup $H$ of $G$ can be generated by at most $r(d)$ elements? ...
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1answer
88 views

Naming: How to call a direct product of elementary abelian groups?

Is there an accepted name for abelian groups of the form $\prod_{i=1}^n \mathbb{Z}_{p_i}$ for some primes $p_1,\dotsc,p_n$? (i.e: direct products of cyclic groups of prime orders, or in other words - ...
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1answer
2k views

Isomorphic quotients by isomorphic normal subgroups

In this recent question, Iota asked if, given a finite group $G$ and two isomorphic normal subgroups $H$ and $K$, it would follow that $G/H$ and $G/K$ are isomorphic. This is not true (a simple ...
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1answer
200 views

Hom functor from the rationals to the rationals

What is $\mathrm{Hom}(\mathbb Q;\mathbb Q)$? Is it isomorphic to $\mathbb Q$? If yes, what is the isomorphism? Is there a reference where the hom functor is explicitly studied with concrete examples ...
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1answer
399 views

Questions on free abelian groups

Is the following true? $G$ is a torsion free abelian group of rank $>n$. Let $S$ be be a subgroup of $G$ generated by $s_1,s_2, \cdots ,s_n$. (1) If $m_1s_1+m_2s_2+ ...+m_ns_n$ is linearly ...
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1answer
249 views

Compact group actions and automatic properness

I am currently re-reading a course on basic algebraic topology, and I am focussing on the parts that I feel I had very little understanding of. There is one exercise in the chapter devoted to groups ...
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2answers
1k views

Isomorphic quotient groups

Suppose that G is a finite group, and that H and K are normal subgroups of G with trivial intersection, and suppose that H and K are isomorphic. Is it true that the quotient groups G/H and G/K are ...
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vote
3answers
276 views

Groups acting on polynomials

I am a little confused about actions right now. Say I have a group $G$ that acts on the roots of a polynomial, say $(x - x_1)(x - x_2)(x - x_3)$. We know that this group permutes the roots of a ...
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2answers
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Property of abelianization

This is related to this old MO question which wasn't answered properly, though I don't feel I phrased the question in the best way (or posted it on the right site) Define the abelianization of a ...