0
votes
1answer
20 views

Conjugating a permutation

I am trying to see that two permutations are conjugate exactly when they have the same cycle decomposition. I fail to see that $$r(i_1,i_2,\dots,i_k)r^{−1}=(r(i_1),r(i_2),\dots,r(i_k))$$ Any thoughts ...
1
vote
1answer
51 views

Derivation and application of Newton's identity

How is the following identity derived? $$\sum_{\ell =0}^{n-1}(-1)^\ell e_\ell s_{n-\ell}+(-1)^nne_n=0$$ Is there an example demonstrating the context in which this might be applied?
2
votes
0answers
23 views

Counting homomorphisms by the order of their images

I am trying to count homomorphisms from $\mathbb Z^r$ to $(\mathbb Z/m)^n$ while keeping track of the order of the image of each map. In other words, for each integer $k$ dividing $m^n$, I want to ...
0
votes
2answers
59 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 ...
7
votes
3answers
181 views

What is the probability of product of two elements is desired element?

Let $G$ be a group with $n$ element. Fix $x\in G$. If you choose randomly two elements from $G$, what is the probability of $x$ being product of these two elements? At first, I thought answer was ...
2
votes
2answers
47 views

How do I calculate permutations where some values are restricted?

I am curious about the formula for determining the number of combinations there are in a given set where some values are restricted to a certain range. For example, if I have a 10 character, ...
1
vote
1answer
139 views

If $|A| > \frac{|G|}{2} $ then $AA = G $

I'v found this proposition. If $G$ is a finite group , $ A \subset G $ a subset and $|A| > \frac{|G|}{2} $ then $AA = G $. Why this is true ?
4
votes
2answers
110 views

How to prove $\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p\;\;$?

An argument that I just came across asserts that if $p$ is prime, and $k, q\in \mathbb{Z}_{>0}$, then $$\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p \;.$$ This assertion was made in a way (i.e. ...
1
vote
2answers
100 views

Find all the elements in $S_4$ that commutes with $(12)(34)$.

Find all the elements in $S_4$ that commutes with $(12)(34)$. And show the algorithm of the process.
5
votes
3answers
97 views

How many Homomorphisms can be between $Z_6$ to $Z_{18}$?

How many Homomorphisms can be between $Z_6$ to $Z_{18}$? and the most important: What is the algorithm for calculating, step by step?
0
votes
2answers
72 views

Compute number of permutations composed only of transpositions for a given set

Given a set of $n$ elements, how can I find the number of all possible permutations composed only by a product of cycles? For example, for the set $\{1,2,3\}$ there are 4 such permutations: $(123)$, ...
2
votes
2answers
91 views

Find the number of permutations in $S_n$ containing fixed elements in one cycle

Find the number of permutations in $S_n$ such as 1 and 2 belong to one cycle.
5
votes
3answers
205 views

Symmetries of combinatorial structures.

Studying the automorphism groups of graphs/finite geometries/designs has been quite useful and important for both group theory and combinatorics. I know of the following books which cover the ideas ...
3
votes
3answers
104 views

Combinatorics inside of $GL(n,q)$

I'm studying conjugacy classes of subgroups of $GL(n,q)$ of the form $(\mathbb{Z}/p\mathbb{Z})^r$ where $q=p^d$ and $r$ is some non-negative integer. I've been able to show that for $n=p=2$ and for ...
0
votes
0answers
34 views

Chromatic classes of vertices of a polyhedron

For a convex polyhedron, how do I figure out all possible proper chromatic classes of its vertices (so that all vertices that are assigned the same color constitute a separate class, and no two ...
0
votes
0answers
53 views

Condition for the number of distinct solutions over GF($q$)

Assume that we have $p$ sets $\left\{ {{m_i}} \right\}_{i = 1}^p$ with given cardinalities $\left\{ {{K_i}} \right\}_{i = 1}^p$, $1 \le {K_i} \le q$, where $q$ is a power of $2$. What I'm trying to do ...
3
votes
1answer
161 views

Definition of the Young Symmetrizer

I have a question regarding the equivalence of two definitions of the young symmetrizer. First, some notation: let $\lambda$ be a partition of $n$. Given a $\lambda$-tableaux $T$, we define the row ...
1
vote
1answer
104 views

Number of Inequivalent Difference Sets In Elementary Abelian 2-groups

I have reason to believe that there is only one$(2^{2s+2},2^{2s+1}-2^s,2^{2s}-2^s)$- difference set (based on experimentation in GAP), up to equivalence/complementation, in any elementary 2-group of ...
3
votes
2answers
49 views

Is there a “natural” way to define a group operation on the set of size-$n$ subsets of a finite set?

It is easy to define a group operation on the set of all subsets of a given finite set S of size n: merely take the exclusive-or (disjoint sum) of the two sets. This is associative, the empty set is ...
6
votes
0answers
252 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
17
votes
1answer
402 views

Probability that two randomly chosen permutations will generate $S_n$.

Every undergraduate learns a fact about the symmetric group that $(1\,2)$ and $(1\,2\,\cdots\,n)$ generate $S_n$. Also, it is interesting to know that for a prime $p$, any 2-cycle and any $p$-cycle ...
8
votes
2answers
224 views

Known bounds for the number of groups of a given order.

The number of nonisomorphic groups of order $n$ is usually called $\nu(n)$. I found a very good survey about the values. $\nu(n)$ is completely known absolutely up to $n=2047$, and for many other ...
1
vote
0answers
50 views

A family of $8$-regular Ramanujan Cayley Graphs

I'm looking for expander graphs with certain properties. Is there a family of $8$-regular Ramanujan Cayley graphs $\{\text{Cay}(G_n,S_n)\}_n$ such that each of them has no cycles of odd length? ...
1
vote
2answers
64 views

Can we get the line graph of the $3D$ cube as a Cayley graph?

Given a graph $G=(V,E)$, the line graph of $G$ is a graph $\Gamma$ whose vertices are $E$ (the edges of $G$) and in $\Gamma$, two vertices $e_1,e_2$ are connected if, as edges in $G$, they share an ...
3
votes
1answer
112 views

Probabilistic Interpretation of Burnside's Lemma

Burnside's Lemma states that $N$, the number of orbits when a group $G$ acts on a set $X$ is given by $$N = \frac{1}{|G|} \sum_{g \in G} |\text{Fix } g|$$ The standard proof involves applying the ...
7
votes
1answer
219 views

Certain Sums of Conjugacy Class Sizes of Symmetric Groups

Suppose $n$ is a natural number and $\lambda$ is an unordered integer partition of $n$ such that $\lambda$ has $a_{\lambda,j}$ parts of size $j$ for each $j$... Let $c_\lambda$ be the conjugacy class ...
10
votes
1answer
227 views

What can we say about the size of $HK\cap KH$ when $HK\neq KH$?

If $G$ is a finite group, and $H$, $K$ are proper subgroups of $G$, then it is not necessary that $HK=KH$. But, these two subsets have same size. The question I would like to ask, then, is If ...
2
votes
0answers
81 views

Find the cycle index of cyclic group $C_p$ where $p$ is prime

Find the cycle index of cyclic group $C_p$ where $p$ is prime. No idea where to start...any help pointing me in the right direction would be appreciated.
7
votes
1answer
209 views

How do combinations (not permutations) relate to group theory?

First question. I'm just generally curious about combinations in group theory. How do they relate? If I take the set of permutations of $\langle 1,2,3,4 \rangle$, I get the symmetry group S4. How ...
5
votes
1answer
509 views

Why is $S_5$ generated by any combination of a transposition and a 5-cycle?

Why is $S_5$ generated by any combination of a transposition and a 5-cycle? Is this true for any prime $p$ (in this case $p=5$)?
2
votes
3answers
50 views

Computing $\langle (13746) \rangle$ in $S_7$.

How to list the elements of subgroup $\langle a \rangle$ in $S^7$ where $$a=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 7 & 6 & 5 & 1 &4 ...
1
vote
5answers
117 views

Permutation question. $qpq^{-1}$, $q,p,r,s \in S_{8}$.

Let $p,r,s,q \in S_{8}$ be the permutation given by the following products of cycles: $$p=(1,4,3,8,2)(1,2)(1,5)$$ $$q=(1,2,3)(4,5,6,8)$$ $$r=(1,2,3,8,7,4,3)(5,6)$$ $$s=(1,3,4)(2,3,5,7)(1,8,4,6)$$ ...
6
votes
2answers
312 views

Involutions and Abelian Groups, II.

In the thread Involutions and Abelian Groups, I supplied a solution to the following interesting problem (with the help of hints provided by the OP). Let $ G $ be a finite group and $ I(G) $ the ...
27
votes
1answer
1k views

Six Frogs - Puzzle

I had come across a puzzle: The six educated frogs in the illustration are trained to reverse their order, so that their numbers shall read 6, 5, 4, 3, 2, 1, with the blank square in its ...
81
votes
1answer
2k views

Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G\: \text{ s.t. } \:H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this ...
7
votes
1answer
541 views

When does the adjacency or incidence matrix of a graph have consecutive ones property?

Given a graph, what are some sufficient (and necessary) conditions to tell if its adjacency matrix has the consecutive ones property? Similar question for its incidence matrix? Note that a ...
62
votes
2answers
4k views

More than 99% of groups of order less than 2000 are of order 1024?

In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024. Is this for real? How can one deduce this result? ...
3
votes
1answer
62 views

Using the 6 sets of sets of 5 disjoint 2-cycles and the 3 sets of 5 pairwise disjoint triangles to exhibit an outer automorphism of $S_6$.

As a follow-up to the construction of a graph that $S_6$ acts on in Constructing a graph based on numbers of vertices, incident edges, and incident triangles, I am now specifically looking at the ...
0
votes
3answers
55 views

Subgroups of $\Bbb Z_n$

Consider the cyclic group $\Bbb Z_n=\{1,2,\cdots n\}$ under addition modulo $n$ and for some non zero $a\in \{1,2,\cdots n-1\}$ let $\langle a\rangle=H\le \Bbb Z_n$ of order $q$. I wish to show that ...
6
votes
1answer
177 views

Uniqueness of conjugates of a subgroup.

This question is partly influenced by the question: Are two subgroups that contain a common element conjugate iff they are conjugate under the normalizer? If we have an arbitrary finite group $G$ ...
5
votes
2answers
128 views

Group with an automorphism of order 2 (Jacobson BA1)

I am having trouble with Exercise 11, Section 1.10 of Basic Algebra 1 by Nathan Jacobson (pub. Freeman & Co. 1985). The statement to prove is: Let $G$ be a finite group and $\phi$ an ...
9
votes
1answer
136 views

Group as product of subsets

There's a fairly simple result that states that for a finite group $G$ and two subsets $A, B$ with $$|A| + |B| > |G|,$$ any $g \in G$ has a representation $g = a*b$ with $a \in A$, $b \in B$. ...
1
vote
1answer
236 views

The sgn function and permutations

Let $P=\{(i, j)|1\leq i<j\leq n\}. $For $\sigma\in S_n$, define $\operatorname{sgn}\colon S_{n}\rightarrow \{\pm 1\}$ by $$ \operatorname{sgn}(\sigma)=\prod_{(i, j)\in ...
1
vote
1answer
43 views

A question about bounding character ratios

The following question arose in a research project, and I'm sure it must be well known. I even know a very indirect proof, and would love to know if anybody knows a simple one. Here is the question. ...
1
vote
0answers
79 views

Sets of independent vectors

So we're working in Z2k, the group of bit-vectors of length k and componentwise addition modulo 2. Now I'm trying to make a function yj=1..?(vi) assigning elements of Z2k to n vertices, such that ...
6
votes
2answers
248 views

What is a natural way to enumerate the symmetries of a cube?

I want to define a labeling, by small natural numbers, of the 48 symmetries of a cube — affine transformations which do not change the volume it occupies. What is a ...
10
votes
2answers
245 views

Finite/Infinite Coxeter Groups

In the same contest as this we got the following problem: We are given a language with only three letters letters $A,B,C$. Two words are equivalent if they can be transformed from one another ...