9
votes
1answer
189 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
33 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
89 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
114 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
41 views
Computing $\langle (13746) \rangle$ in $S_7$.
How to list the elements of subgroup $\langle a \rangle$ in $S^7$ where $$a=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 7 & 6 & 5 & 1 &4 ...
6
votes
2answers
90 views
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
Assume that $\pi$ is an isomorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. Anyone ...
2
votes
5answers
84 views
Permutation question. $qpq^{-1}$, $q,p,r,s \in S_{8}$.
Let $p,r,s,q \in S_{8}$ be the permutation given by the following products of cycles:
$$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
131 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 ...
22
votes
1answer
430 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 ...
45
votes
0answers
828 views
+50
Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,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 ...
6
votes
1answer
299 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 ...
48
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
51 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
54 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
157 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$ ...
6
votes
2answers
96 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
85 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|,$$ any $g \in G$ has a representation $g = a*b$ with $a \in A$, $b \in B$. ...
1
vote
1answer
199 views
The sgn function and permutations
Let $P=\{(i, j)|1\leq i<j\leq n\}. $For $\sigma\in S_n$, define $\operatorname{sgn}\colon S_{n}\rightarrow \{\pm 1\}$ by $$
\operatorname{sgn}(\sigma)=\prod_{(i, j)\in ...
1
vote
1answer
41 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
75 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
179 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
193 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 ...
