The study of symmetry: groups, subgroups, homomorphisms, group actions.
5
votes
0answers
48 views
Why are the p-adic integers a linearly ordered group? [duplicate]
In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group.
I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
4
votes
2answers
93 views
Finite abelian $2$-group
If $G$ is a finite abelian group such that $o(x)=2$ for all $x \neq e$ and $|G|=2^n$ for some $n\in\mathbb N$, prove that $G \cong \mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ ($n$ factors).
Any ...
1
vote
1answer
52 views
Set of left cosets is a group
Let $N$ be a normal subgroup of $G$. Let $G/N = \{gN : g \in G \}$. Show that the $(G/N, \circ, 1_{G/N})$ is a group, where:
$\circ : G/N \times G/N \rightarrow G/N$ is defined by $(gN \circ hN) = ...
1
vote
0answers
46 views
Explanation of notations
I was reading Binary icosahedral group in Wikipedia. The author uses $<2,3,5>$ and $(2,3,5)$ to denote the groups. And the Coxeter group of type $H_4$ is denoted by $[3,3,5]$. Could anyone ...
2
votes
1answer
47 views
Understanding the internal direct product of a group.
I have come across the statement:
$G$ is the direct product of its subgroups $N_1$ and $N_2$ if the following conditions hold:
1) $N_1,N_2$ are normal subgroups.
2) $N_1\cap N_2=\{e\}$
3) They ...
8
votes
2answers
35 views
The index of $\xi_4^*$ in $\xi_4$
Just seeing if i'm right:
With the set of solutions for $z^4=1$: $\xi_4=\{1,i,-1,-i\}$, one can construct the group of the $4$th roots of unity: $(\xi_4,\cdot_\mathbb{C})$ and its multiplicative ...
3
votes
1answer
28 views
Show that the subgroup $s(K) \subset H \rtimes_{\alpha} K$ is normal if and only if $\alpha: K \rightarrow Aut(H)$ is the trivial homomorphism.
Show that the subgroup $s(K) \subset H \rtimes_{\alpha} K$ is normal if and only if $\alpha: K \rightarrow Aut(H)$ is the trivial homomorphism, where $s : K \rightarrow H \rtimes_{\alpha} K$ is given ...
0
votes
2answers
52 views
Rings | Homomorphisms | Units
Question
Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is
a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$.
Attempt
...
2
votes
1answer
98 views
What is the reason for the name *left* coset?
Let $G$ be a group and let $H \leq G$ be a subgroup. It seems that it is now standard to call the cosets
$$gH=\{gh \ | h \in H \}$$ the left cosets of $H$ in $G$. I have to admit to being slightly ...
0
votes
1answer
44 views
ordering of a group
An ordering of a group $G$ is a linear ordering $<$ on (the underlying set of) $G$
that satisfies, in addition,
$a < b \implies ac < bc ∧ ca < cb$,
for all $a,b, c \in G$.
Show that a ...
2
votes
2answers
38 views
On the eigenvalues of a linear transformation $\tau$ such that $\tau^3 = \mathrm{id}$
I am reading the book on representation theory by Fulton and Harris in GTM. I came across this paragraph.
[..] we will start our analysis of an arbitrary representation $W$ of $S_3$ by looking ...
4
votes
1answer
22 views
Proving that the Flag Variety $Fl(n;m_1,m_2)$ is connected.
I wish to prove that the flag variety $Fl(n;m_1,m_2) = \{ W_1 \subset W_2 \subset V | dimW_i = m_i \}$, for $0 \le m_1 \le m_2 \le n$ where V is an n-dimensional vector space over $\mathbb{C}$ and ...
6
votes
1answer
45 views
Given $G$, when can we find a division ring $R$ with $R^*=G$?
This is motivated by a characterization of finite cyclic groups, in which one proves
Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic.
The proof is ...
2
votes
2answers
58 views
question on subgroups of prime order
Let $G$ be a group and let $\,H,\, K\,$ be subgroups of $\,G,\,$ each of order $\,p,\,$ where $\,p\,$ is prime.
Show that either $\,H\cap K =\{e\},\,$ or $\,H=K.\,$
Is the result true if ...
-3
votes
0answers
39 views
Show there does not exist $\alpha$ in $ S_4$ s.t. $ (1 2)(3 4)\alpha = \alpha(1 2 3 4)$ [duplicate]
Show that there does not exist a permutation $\alpha \in S_4$ such that $ (1 2)(3 4)\alpha = \alpha(1 2 3 4)$.
1
vote
0answers
33 views
A strange characterisation of cyclic groups [duplicate]
"A finite group is cyclic if, for any integer m, the number of elements
of order dividing m is at most m."
I have never seen this characterisation of cyclic groups before. How do I prove this? I hope ...
3
votes
1answer
46 views
sylow basis of finite solvable groups
Let $G$ be a finite solvable non-$p$-group and $A$ be a maximal subgroup of $G$.
Therefore $A$ is of primary index $p^{n} $, that is $|G : A|=p^{n}$ where ...
1
vote
3answers
88 views
Prove $rs=sr^{-1}$ in ${\rm Dih}(2n)$
Let $r$ and $s$ be the rotation and reflection symmetries respectively in ${\rm Dih}(2n)$, the dihedral group of order $2n$. Show that $rs=sr^{-1}$.
I also need to show by induction that ...
0
votes
1answer
56 views
E measurable with m(E) < $\infty$?
Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$.
ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable.
I told my ...
3
votes
0answers
50 views
Can all non-archimedean groups be written as a product of archimedean groups?
All the non-archimedean groups I know of can be written as the product of archimedean groups. I'm wondering if this is generally true. I think I've found a proof, but I haven't heard this theorem ...
1
vote
2answers
46 views
The set of complex numbers of modulus $1$ is a group under multiplication
Show that $C=\{z\in \mathbb{C} \mid |z|=1\}$ is a group under complex multiplication.
I'm a little confused because isn't the identity the only element with order $1$? What is this set?
4
votes
2answers
62 views
Cardinality of $GL_n(K)$ when $K$ is finite
I don't know how to do the last task of an exercise.
Let $K$ be a field, $G=GL_n(K)$ and $X=K^n\backslash\{0\}$.
First task: Show that $G \times X \to X$, $(A,x)\mapsto Ax$ defines an action of $G$ ...
1
vote
1answer
60 views
Non trivial Automorphism [duplicate]
Prove that every finite group having more than two elements has a nontrivial Automorphism.
It is from Topics in Algebra by Herstein. I am not able to solve.
2
votes
2answers
40 views
On the proof of Schur's lemma in Fulton & Harris
I'm reading the book on representation theory by Fulton and Harris. I'm stuck with the proof of Schur's Lemma 1.7:
Schur's lemma 1.7 If $V$ and $W$ are irreducible representation of $G$ and ...
6
votes
3answers
78 views
Finding the subgroup of $(\mathbb Z_{56},+)$ which is isomorphic with $(\mathbb Z_{14},+)$ by GAP
I am sorting some easy questions for the students in Group Theory I. One of them is:
Is $(\mathbb Z_{14},+)$ isomorphic to a subgroup of $(\mathbb Z_{35},+)$? What about $(\mathbb Z_{56},+)$?
I ...
4
votes
3answers
80 views
$\mathbb{Q}/\mathbb{Z}$ has cyclic subgroup of every positive integer $n$? [duplicate]
I would like to know whether $(\mathbb{Q}/\mathbb{Z},+)$ has
$1$. Cyclic subgroup of every positive integer $n$?
$2$. Yes, unique one.
$3$. Yes, but not necessarily unique one.
$4$. Does not have ...
2
votes
0answers
41 views
Universal coefficient theorem maps
Let $G$ be a group and $A$ a trivial $G$-module. The Universal Coefficient Theorem yields a split exact sequence
$$0\longrightarrow {\rm Ext}(H_{n-1}(G),A)\longrightarrow H^n(G,A)\longrightarrow {\rm ...
5
votes
0answers
49 views
+100
Amenable group rings embeddable in skew fields
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent:
(1) the group ring $K[G]$ is a domain;
(2) $K[G]$ is ...
5
votes
3answers
98 views
Homomorphism from $\mathbb{Z}/n\mathbb{Z}$
Does there exist a homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly?
4
votes
3answers
65 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
23 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
29 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
69 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
64 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
87 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
64 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
50 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
59 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
60 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 ...
