The study of symmetry: groups, subgroups, homomorphisms, group actions.
4
votes
3answers
66 views
Homomorphisms between $ \mathbb{Z} $ modules.
Calculate $\newcommand\Hom{\operatorname{Hom}}\Hom(\mathbb Z \oplus \mathbb Z_{p^\infty},\mathbb Z \oplus \mathbb Z_{p^\infty})$. Where $ \mathbb Z_{p^\infty}= ...
2
votes
0answers
26 views
Extending transvections/generating the symplectic group
The context is showing that the symplectic group is generated by symplectic transvections.
At the very bottom of http://www-math.mit.edu/~dav/sympgen.pdf it is stated that any transvection on the ...
1
vote
1answer
39 views
Determine no. elements in $ \operatorname{Aut}(H)$ where $H$ is the 6 point/5line graph in the shape of H
Determine the amount of automorphisms in the group $\operatorname{Aut}(H)$ where $H$ is the graph with 6 points and five lines in the shape of a capital 'H'. Here is what it should look like, I ...
-1
votes
0answers
55 views
Orbits of group actions
I have the following problem:
Describe the orbits of the group action in each of the following cases (you are not asked to show they are actions):
(a) $(0,\infty)$ acts on C by multiplication
(b) ...
0
votes
0answers
31 views
conjugacy class sums in group rings
Let $G$ be a finite group and
let $\mathbb{Q}[G]$ be its rational group ring.
It is known that the conjugacy class sums form a basis
for the center of $\mathbb{Q}[G]$ so that there cannot ...
3
votes
1answer
35 views
Determine the number of elements of order 2 in AR
So i have completed parts a and b. For b i reduced R to smith normal form and ended up with diagonals 1,2,6. From this i have said that the structure of the group is $Z_2 \oplus Z_6 \oplus Z$. But i ...
4
votes
1answer
70 views
Group actions and natural isomorphisms
Let $G$ be a group-as-category, and let $S$ be the image of $G$ by the $Hom(G,-)$ functor. The $Hom$ functor defines a bijection $\chi: G \to S$ between elements of $S$ and morphisms of $G$, and thus ...
4
votes
3answers
77 views
on the commutator subgroup of a special group
Let $G'$ be the commutator subgroup of a group $G$ and $G^*=\langle g^{-1}\alpha(g)\mid g\in G, \alpha\in Aut(G)\rangle$.
We know that always $G'\leq G^*$.
It is clear that if $Inn(G)=Aut(G)$, then ...
2
votes
2answers
48 views
Direct Product of the $G_i $'s
I am a little confused in the interpretation of the product of groups. Here is what's written in my notes:
Given groups $G_1,G_2,...,G_n$, recall that $G_1\times G_2\times ...\times ...
0
votes
0answers
22 views
Describe the symmetry group D of T of all rotations and reflections in R^2..
Describe the symmetry group D of T of all rotations and reflections in $R^2$ as a subgroup of the symmetric group $\Sigma_5$. Where T is the regular pentagon (5-gon), with vertices enumerated ...
3
votes
4answers
66 views
How to find all elements of $\mathbb{Z}_{4} \times\mathbb{Z}_{4}/\langle(1,1)\rangle$?
I am studying factor groups, and I saw an example that says
Find all the elements of the factor group $\mathbb{Z}_{4} \times \mathbb{Z}_{4}/\langle(1,1)\rangle$.
I know that the order of ...
4
votes
2answers
65 views
Which one of the following groups is decomposable?
A group $(G,+)$ is said to be decomposable if $G$ has two non-trivial subgroups $G_1$ and $G_2$ such that $G=G_1+G_2$ and $G_1 \cap G_2 =$ {$e$}. Then which of the following are decomposable:
(i) ...
2
votes
3answers
89 views
Why can't this be a coset?
Let $H$ be a subgroup of $G$ and H is not normal, there are left cosets $aH$ and $bH$ whose product isn't a coset.
My attempt:
$ab H\subset aHbH$ and if H is not normal, if $ah_1bh_2=abh_3$ for all ...
6
votes
2answers
66 views
Embeddings of $GL(n-1,q)$ into $GL(n,q)$
Let $n>2$ and $q$ be a prime power. I want to find all the embeddings of $GL(n-1,q)$ into $GL(n,q)$. An embedding I first thought of is as following. Let $V$ be an $n$-dimensional vector space over ...
2
votes
2answers
48 views
Rotman's Introduction to to the theory of groups. Exercise 3.45.
Can you give me a hint on the first part of the exercise?
Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
4
votes
1answer
65 views
When are groups of order 12 non-abelian?
I am currently reading http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/group12.pdf, and have a quick question about the group being non-abelian. Let me explain:
Let $|G|=12=2^2\cdot 3$ and let ...
2
votes
1answer
83 views
What does it mean to “Decide to which group $G$ is isomorphic” for a given group $G$?
I have a homework question which is
Decide to which group $(\mathbb{Z}_n^*,\,\cdot\,)$ is isomorphic (classification of finite abelian groups), for
(i) $n = 9$,
(ii) ...
4
votes
1answer
53 views
Order of this group
Given $k\in{\mathbb{N}}$, we denote $\Gamma _2(p^k)$ the multiplicative group of all matrix $\begin{bmatrix}{a}&{b}\\{c}&{d}\end{bmatrix}$ with $a,b,c,d\in\mathbb{Z}$, $ad-bc = 1$, $a$ and $d$ ...
1
vote
1answer
51 views
Explain the formula for the size of an orbit…
Explain the formula for the size of an orbit and show that this always is a divisor of the group order $|G|$. (You may use Lagrange's theorem!)
So I would like to know how i can go about answering ...
8
votes
1answer
60 views
Cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$?
What are the cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$, the general linear group over the finite field of order $p$, where $p$ is prime?
Obviously, each cyclic subgroup is generated by some ...
2
votes
2answers
61 views
A mathematical model for rotations of a die
I have a normal 6-sided die and I would like to find a model for how it transforms under rotatins (north, south, east, west), that can help me determine which side is up. Just to make sure we are ...
4
votes
3answers
41 views
Automorphism of $K$ extending to $K[x_1,\dots,x_n]$.
I think it is well known that if $K$ is field, and $\sigma \in Aut(K)$,
then the extended map $\sigma: K[x,y] \rightarrow K[x,y]$, given by
$\sum a_{ij}x^iy^j \mapsto \sum \sigma(a_{ij})x^iy^j$ is ...
1
vote
1answer
58 views
Let $G$ be a set with associative binary operation and a unit.
Let $G$ be a set with associative binary operation and a unit. Assume that for every $ g \in G$ there exists $ x \in G$ with $xg = 1$. Prove that $gx = 1$ is a consequence.
That above is the ...
0
votes
1answer
62 views
1
vote
2answers
75 views
$H$ must contain every Sylow $p$-subgroup of $G$
G is finite and has a prime factory. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that
4
votes
2answers
60 views
If the order of a finite abelian group is not divisible by a square, show that the group must be cyclic.
If the order of a finite abelian group is square free, show that the group is cyclic.
This is a question from "basic abstract algebra" by bhattacharya
3
votes
2answers
62 views
Silliness: $\exists~X~\text{s.t.}~AX=B \iff B\in R(L_A)$
So, I am asked to prove that the system of linear equations $AX=B$ has $\color{black}{a~solution}$ if and only if $B\in R(L_A)$. $R$ denotes the "range of" and $L_A$ is left multiplication by $A$. If ...
7
votes
1answer
73 views
Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that the order of the center of $G$ is 1 or $pq$.
Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that
the order of the center of $G$ is 1 or $pq$.
Let me start off with what I did:
Assume $G$ is abelian. Then we know ...
2
votes
2answers
49 views
Prove that a subgroup which contains half of all elements is a normal subgroup.
I have been going through some material on elementary group theory (Group Theory for Physicists, Christoph Ludeling) and in one place it states:
A subgroup which contains half of all elements, i.e. ...
0
votes
1answer
42 views
Discuss whether or not the following binary operations are commutative, associative, …
Discuss whether or not the following binary operations are commutative, associtive, have neutral elements and for which elements there are inverse elements.
In between are what I have said, but if it ...
3
votes
3answers
77 views
Sum of two squares in a $\Bbb Z/p\Bbb Z$
I need to show that every element in $\Bbb Z/p\Bbb Z$ can be written as a sum of two squares. The case $p=2$ is trivial and $0$ is always $0^2 + 0^2$. So all I have to do is show that every element of ...
1
vote
1answer
38 views
$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?
In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
5
votes
3answers
102 views
What is the meaning of the parentheses in $\phi^{-1}\left[\{\phi(g)\}\right]=gH=Hg$?
I am studying homomorphisms is groups and i saw a theorem saying:
For $g$ in a group $G$, the cosets $gH$ and $Hg$ are the same, and collapsed onto the single element $\phi(g)$ by $\phi$. That is, ...
2
votes
1answer
90 views
A question about basis linear space
See $\mathbb{R}$ as a linear space over $\mathbb{Q}$. $B\subset\mathbb{R}$ is a basis with positive reals.
Can $B$ be a group where multiplication is standard?
5
votes
2answers
56 views
Subgroup transitive on the subset with same cardinality
Maybe there is some very obvious insight that i miss here, but i've asked this question also to other people and nothing meaningful came out:
If you have a subgroup G of $S_n$(the symmetric group on ...
0
votes
1answer
48 views
Groups and automorphisms of group-1
Why valid results don't understand for groups $V$ seen as automorphisms of group
$G$ is also valid when it is considered those groups as subgrupos of $G$.
For instance, the following result is true:
...
3
votes
1answer
72 views
Historical definition of a group
Wikipedia states that van Dyck (1882) was the first to give the definition of a group in the modern way. Before this, what were some of the original axioms or conditions for groups? I mean, how were ...
0
votes
0answers
47 views
Generators in $p$-groups
Let $G$ be a non-abelian finite $p$-group such that $G/[G,G]$ is elementary abelian group of order at least $p^3$. Let $S=\{\bar{x_1},\bar{x_2},\bar{x}_3,\cdots,\bar{x}_n\}$ be independent generators ...
7
votes
1answer
66 views
Why does the automorphism used to construct the group have to be non-inner?
I have a question on why a particular assumption is made that the automorphism used to construct a certain group be non-inner.
In [Herstein, Topics in Algebra, p. 69], a construction of a nonabelian ...
1
vote
2answers
74 views
Is $\operatorname{GL}(n, \mathbb{R})$ with multiplication a group?
I am looking at an exercise that saying $GL(n,\mathbb{R})$ with multiplication, in other words the nxn matrices with real entries together with multiplication is a group. I wonder the following: do ...
0
votes
0answers
26 views
An question of $\pi$-groups for the alternating group $A_5$
Let $G=A_5$, the alternating group of degree 5. Then for $\pi = \{2,3\}$ we have that $M \leq G$ is a maximal $\pi$-group if, and only if $M \cong A_4$ or $M\cong S_3$ (the symmetric group).
0
votes
0answers
42 views
Solvable groups of order less than 168
I am asking myself the following question:
What are the solvable groups of order less than 168 ?
First of all, the smallest non-abelian simple group is $A_5$ (order $60$). Let $H$ be a group of ...
1
vote
2answers
90 views
Multiplicative group of integers modulo n
Consider the abelian group $U_n=\{a\in \mathbb{Z}_n:(a,n)=1\}$. Is there a natural way to understand it as a subgroup of any other interpretation of the cyclic group of order $n$. For example, ...
3
votes
1answer
47 views
Sufficient condition for a direct limit of abelian groups to be infinitely generated
I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
3
votes
1answer
53 views
The group $\mathbb Q^*$ as a direct product/sum
Does the group $\mathbb Q^*$ (rationals without $0$ under multiplication) is a direct product or a direct sum of nontrivial subgroups?
My thoughts. Consider subgroups $\langle p\rangle=\{p^k\mid k\in ...
-1
votes
1answer
61 views
Generators of a cyclic group and their orders
a) Let $G = \langle a \rangle$ be a finite cyclic group. Prove that for each $b\in G$, $\langle b \rangle=G$ if and only if order of $b$ equals order of $G$.
b) The previous part does not hold if $G$ ...
3
votes
0answers
72 views
How to recover the integral group ring?
I would like to solve the following exercise:
Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times ...
3
votes
2answers
130 views
Group theory between mathematicians and physicists?
What is the difference between the study of group theory as a mathematical subject and as a physical method in quantum mechanics, for example?
When a person studies group theory, what subjects does ...
3
votes
1answer
54 views
Counting 0-1 matrices up to symmetry
I'm interested in counting the number of n×n 0-1 matrices with a given number of 1s up to rotation and reflection. What is the best way to do this if n is not too small?
For example, consider ...
3
votes
0answers
42 views
Any finite subgroup of the abelian group $(F-\{0\},\cdot)$ is cyclic? ($F$ a field) [duplicate]
I found this problem:
Suppose that $F$ is a field, and that $(F-\{0\},\cdot)$ is an abelian group. Show that if $H$ is a finite subgroup of $F-\{0\}$, then $H$ is cyclic.
What I have done is:
...
