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

learn more… | top users | synonyms (2)

1
vote
0answers
74 views

Translate group definition into geometry system

I need to reword the definition of group (the four axioms: closure, associativity, identity and invertibility) to be lines and points of non-Euclidean geometry (the axiom system defined as geometry). ...
2
votes
0answers
112 views

What is the easiest example of a finitely presented group which is not residually finite?

What is an easy example of a finitely presented group which is not residually finite? To be clear, part of the question is: how do we see that it isn't?
5
votes
1answer
110 views

Let $A$ be a symmetric subset of a group $G$ such that $A$ contains the identity, and $A$ is covered by some translation. Is then $A$ a subgroup?

Let $G$ be a multiplicative group and $A\subseteq G$ such that 1) $\forall a\in A, a^{-1}\in A$ 2) $1\in A$ 3) $AA \subseteq gA$ for some $g\in G$ Can we say that $A$ is a subgroup? One can ...
7
votes
1answer
779 views

order of element in symmetric group

let $n=p_1+p_2+\cdots+p_k$ ($p_k$ is kth prime number) then $\prod_{i=1}^k p_i$ is maximum order in $S_n$. I think it is easy but I am trying to prove it , but I have not any idea how to deal with ...
3
votes
1answer
118 views

Am I making sense with this argument concerning Symmetric groups?

If $p$ is a prime number, then am I right in thinking that there is only one order $p$ subgroup in the symmetric group $\operatorname{Sym}(p)$? My rationalization is as follows, please correct me if i ...
0
votes
2answers
220 views

Normal subgroups

What are the usual tricks is proving that a group is not simple? (Perhaps a link to a list?) Also, I may well be being stupid, but why if the number of Sylow p groups $n_p=1$ then we have a normal ...
11
votes
2answers
3k views

Is there a systematic way of finding the conjugacy class and/or centralizer of an element?

Is there a systematic way of finding the conjugacy class and centralizer of an element? Could the task be simplified if we are working with "special groups" such as $S_n$ or $A_n$? Are there any ...
1
vote
1answer
302 views

An equivalent definition of a group

Exercise 15 from Hungerford: Algebra. Let $G$ be a nonempty finite set with an associative binary operation such that for all $a,b,c\in G\,\,ab=ac \Rightarrow b=c$ and $ba=ca \Rightarrow b=c$. Then ...
4
votes
1answer
241 views

When is the image of a group morphism a normal subgroup?

Let $f : G \to G'$ be a group morphism. I need to find a necessary and sufficient condition such that $\operatorname{Im}(f)$ is a normal subgroup of $G'$.
2
votes
2answers
341 views

Symmetric Group S3 Symmetry

Consider the action of the full symmetric group $S_3$ on the cube $[0,2] \times [0,2] \times [0,2]$. Classify the orbits of this action and determine their cardinalities. My Answer: What I note is ...
4
votes
1answer
204 views

Fermat's little theorem for $n=3$

for $N > 0$, I'm trying to show Fermat's little theorem, for $3$ using the orbit stabilizer theorem: $N^3 - N$ an element of $3\mathbb{Z}\ (3 \mod \mathbb{Z})$ Pf/ we can break it down into ...
3
votes
1answer
271 views

How to calculate inverses in the group algebra

Is there an algorithm to calculate the inverse of an element in the group algebra? For example, does the element $(1 2 3) + 2 . (1 2)(3 4)$ in the group algebra $\mathbb{C} S_{4}$ have an inverse, ...
2
votes
2answers
680 views

Cyclic Groups, Quotient Groups [duplicate]

Possible Duplicate: $G$ modulo $N$ is a cyclic group when $G$ is cyclic Prove that if $H$ is a subgroup of a cyclic group $G$, then $G/H$ must also be cyclic. I think that I start off ...
5
votes
2answers
177 views

Why do these two presentations present the same group?

Could you explain to me why these two presentations present the same group? $$\begin{align*} &\bigl\langle a,b\ :\ aba=bab\bigr\rangle\\ &\bigl\langle x,y\ :\ x^2=y^3\bigr\rangle ...
2
votes
2answers
233 views

A problem concerning action via automorphisms

This problem asks to show that if $A$ acts on $G$ via automorphisms, where either $A$ or $G$ is soluble and both $A$ and $G$ are finite groups and $G$ is nontrivial, then $G$ possesses an ...
2
votes
2answers
166 views

$A_5$ problem with normal subgroup

Let G=$A_5$ and $H=\bigl\langle (12)(34),(13)(24)\bigr\rangle$. Prove $(123) \in N_{G}(H)$ and hence deduce the order of $N_{G}(H)$. I know you claim that $A_5$ is simple, then $N_{G}(H)$ has ...
5
votes
1answer
461 views

Proof of no simple group of order 992

Prove there are no simple groups of order $992$. Factorise it. $31 \times 2^5 $ so you have $|G|=31 \times 2^5 \geq n_{31}(31-1)+ n_{2}(2^5-1)+1$ Putting it in Sylow theorem. So how do you get ...
2
votes
1answer
350 views

Proof that $A_5$ is simple

The thing I don't understand about the proof is that we are using the fact that $A_5$ contains all it's three cycles. I can't see why this doesn't prove $A_4$ is simple? How can you use this argument ...
2
votes
1answer
163 views

Showing $A_4$ is not simple.

This is meant to be a counter example to $A_4$ being simple, since $\{ (1),(12)(34),(13)(24),(14)(23)\}$ is normal. But, how can you check it's normal, is there a quick way or do you need to ...
2
votes
0answers
56 views

For a $G$-module $A$ is there a maximal subgroup $H$ of $G$ such that the image $H^2(G,A)\rightarrow H^2(H,A)=0$?

Let $G$ be a group, and let $A$ be a $G$-module. Then for every subgroup $H$ of $G$, $A$ is also an $H$-module. Furthermore, there's a map $H^2(G,A)\rightarrow H^2(H,A)$. I would like to know ...
53
votes
6answers
2k views

Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me ...
28
votes
3answers
973 views

What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the ...
3
votes
1answer
118 views

Is the rank of a relatively free group… ill-defined in general?

A relatively free algebra $F$ has a free generating set (basis) $X$ such that any map $f : X \to F$ can be extended to an endomorphism of $F$. It is known that, in general the notion of rank of $F$ ...
2
votes
3answers
3k views

example for non-abelian group

We defined a group as a set $G$ with an operation $\circ$ with 1) $\forall x,y,z, \in G: x \circ ( y \circ z) = (x \circ y) \circ z$ 2) $\exists e\in G: \forall x\in G : x \circ e = e \circ x = x$ ...
2
votes
3answers
261 views

Existence of a subgroup of order $pq$

Let $G$ be a finite group and assume $G$ has a single $p$-Sylow subgroup. Let $q\neq p$ be prime for which $q$ divides the order of $G$. Show that there exists a subgroup of $G$ with order $pq$. ...
4
votes
1answer
45 views

Is the number of index d subgroups in the free group of rank 2 bounded by a polynomial?

For $d=1$, let $M_1 = 1$. For $d>1$, define $M_d$ recursively by $$M_d = d(d!) - \sum_{i=1}^{d-1} (d-i)! M_i.$$ Is $M_d$ bounded by a polynomial (of some high degree) in $d$? Note that $M_d$ is ...
3
votes
4answers
2k views

How to prove that the converse of Lagrange's theorem is not true?

I consider the Lagrange theorem. Let $G$ be a finite group and let $H \subseteq G$ be a subgroup, then the order of $H$ divides the order of $G$. I am interesting with the proof of this theorem. The ...
1
vote
1answer
144 views

Characterization of a Normal Subgroup

I am studying a proof of a theorem that states that a subgroup $N$ of a group $G$ is normal in $G$ if and only if $(xA)(yA) = (xy)A$ for all $x,y \in G$. The author goes through a fairly involved ...
4
votes
1answer
174 views

Burnside theorem?

Burnsides theorem If $G$ has $t$ orbits on $\Omega$, then $t=\frac{1}{|G|} \sum_{g \in G} |\operatorname{fix}_{\Omega} (g)|$ It seems to be done by counting in two ways , then saying that the ...
6
votes
3answers
1k views

If $|G| = pq$, then $G$ has an element of order $q$

Suppose I have a group $G$ order $pq$, where $p$ and $q$ are distinct primes, and I know that there exists $a \in G$ such that $order(a) = p$. How do I show that there exists $b \in G$ such that ...
9
votes
0answers
274 views

On group automorphism of subgroup a group $G$

Let $G$ be a group and $H$ be a subgroup of $G$. When is $\rm{Aut}(H)$ a subgroup of $\rm{Aut}(G)$?
7
votes
2answers
998 views

A Nontrivial Subgroup of a Solvable Group

Question: Let $G$ be a solvable group, and let $H$ be a nontrivial normal subgroup of $G$. Prove that there exists a nontrivial subgroup $A$ of $H$ that is Abelian and normal in $G$. [ref: this is ...
0
votes
3answers
191 views

Word problem in a free group

Can the word problem in a free group be solved by a finite state automaton? I know it can be solved by a pushdown automaton.
1
vote
1answer
177 views

Does there exist an abelian $2$-group of finite exponent that is not a direct sum of cyclic groups?

Does there exist an abelian $2$-group (an abelian group, all of whose elements have order a power $2$) of finite exponent that is not isomorphic to a direct sum of $2$-cyclic groups? The exponent of ...
3
votes
2answers
669 views

If $K$ is a subgroup of $G$, then $\phi(K) = \{ \phi(k) | k \in K \}$ is a subgroup of $\bar{G}.$ Vice-versa?

Suppose $\phi: G \to \bar{G}$ is an isomorphism from one group to the other. Then the following is true: If $K$ is a subgroup of $G$, then $\phi(K) = \{ \phi(k) | k \in K \}$ is a subgroup of ...
0
votes
0answers
113 views

How can I generate $\mathrm{SL}(n,\mathbb Z)$ by the subgroup $\mathrm{SL}(n-1,\mathbb Z)$ and another Element of $\mathrm{SL}(n,\mathbb Z)$?

Let $\{z_1,...,z_n\}$ be the canonical Basis of $\mathbb{Z}^n$, such that $z_i$ equals the vector $(0,\dotsc,0,i,0,\dotsc,0)$ with a 1 in the $i$th position. I want to show that the ...
2
votes
1answer
189 views

$k$-transitive group actions for $k>5$

I came across the following claim in "Adventures in Group Theory" by David Joyner that I couldn't find a proof for. If $k>5$ and $G$ is a group acting $k$-transitively on a finite set $X$ then $G$ ...
0
votes
1answer
211 views

Is a subgroup of a direct sum of cyclic finite $p$-groups also a direct sum of cyclic finite $p$-groups?

Let $G = \sum_{i\in I}H_i$ where $H_i$ are finite cyclic $p$-groups and $I$ may be infinite. Let $T$ be a subgroup of $G$. Is it true that $T = \sum_{i\in I}N_{i}$ where $N_i$ is normal subgroup of ...
2
votes
1answer
246 views

Rank of a transitive G-set and double cosets

This is a follow up question to a previous question: If $X$ is a transitive $G$-set, then the rank of $X$ is the number of $G_x$-orbits of $X$ where $G_x$ is the stabilizer of some $x\in X$. If $X$ ...
2
votes
1answer
186 views

Is this true? Every abelian p-group is isomorphic some direct sum of cyclic p groups.

Every abelian $p$-group is isomorphic to a direct sum of cyclic $p$-groups. We have that every abelian $p$-group is an image of some direct sum of cyclic $p$- groups. Therefore, every abelian ...
3
votes
0answers
298 views

How to generate an $n \times n$ rotation matrix?

It is well known that the $2 \times 2$ rotation matrix is given by, $$\left[ \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array} \right]$$ and ...
3
votes
3answers
252 views

Which Digit-Permutations Preserve Divisibility?

This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...
5
votes
3answers
326 views

Intersection of a Subgroup with an Abelian Subgroup is Normal in the Product

Question: Let $A$ be an Abelian group with $A \trianglelefteq G$, and let $B \leq G$ be any subgroup. Show that $A \cap B \trianglelefteq AB$. [ref: this is exercise 20 on page 96 of [DF] := Dummit ...
0
votes
2answers
111 views

Characterizations of rank of a transitive G-set

If $X$ is a transitive $G$-set, then the rank of $X$ is the number of $G_x$-orbits of $X$ where $G_x$ is the stabilizer of some $x\in X$. "An Introduction to the Theory of Groups" by Joseph Rotman ...
4
votes
2answers
122 views

Commutator of a particular group product

Let $S \leq G$ and $N \lhd G$ two subgroups of $G$. Is the commutator $[SN,SN]$ equal to $[S,S]N$ ? The first is generated by commutators $[ax,by]$ (where $a,b\in S$ and $x,y \in N$), and each ...
4
votes
1answer
102 views

Why does every finite subgroup of $\mathrm{Aut}(F_n)$ acts on a graph of Euler characteristic $n-1$?

My question is the following: In a paper I read that: Any finite subgroup of $\mathrm{Aut}(F_n)$ can be realised as agroup of baspoint-preserving isometries of a graph of Euler characteristic $1-n$. ...
8
votes
1answer
422 views

Number of elements in a group of different orders, mod $n$, yields subgroup information?

The alternating group on five elements, $A_5$, has $1$ element order $1$, $15$ elements order $2$, $20$ elements order $3$, and $24$ elements order $5$. If I take these numbers mod some $n$, sometimes ...
3
votes
1answer
757 views

Intersection of subgroups of orders 3 and 5 is the identity

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
3
votes
1answer
483 views

Cosets of a subgroup do not overlap

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
2
votes
0answers
123 views

Hirsch length of polycyclic groups

I've got the following exercise to solve: Let $G$ be a polycyclic group, $N \lhd G$ and let $h(\cdot)$ be the Hirsch length. Then $h(G) = h(N) + h(G/N)$ I know that subgroups and quotients of ...