The study of symmetry: groups, subgroups, homomorphisms, group actions.
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 ...
2
votes
1answer
42 views
On the Cyclic Subgroups
I recently read in a book on group theory that, a (non-abelian) $2$-generator group all of whose proper subgroups are cyclic, has been constructed by Ol`sanski; these are infinite simple groups. (see ...
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 ...
2
votes
1answer
35 views
Minimal Polynomials Annihilating an Abelian Torsion-Free Group
Let $A$ be an abelian torsion-free group. Let $\theta \in\operatorname{Aut}A$. Assume that $\theta$ has a finite period in $\operatorname{Aut} A$, say $n$. Obviously $\theta^n-1$ annihilates $A$ (i.e. ...
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 ...
2
votes
3answers
86 views
All group homomorphism from $ \mathbb{Z} _m $ to $\mathbb{Z}_n $ [duplicate]
All group homomorphism from $ \mathbb{Z} _m $ to $ \mathbb{Z}_n $
How could I find every group homomorphism?
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 ...
4
votes
2answers
1k views
A normal subgroup intersects the center of the $p$-group nontrivially
If $G$ is a finite $p$-group with a nontrivial normal subgroup $H$, then the intersection of $H$ and the center of $G$ is not trivial.
2
votes
1answer
123 views
Commutative ring with unity Proof on the set of units?
the question is as follows
(TRUE or FALSE.) If R is a commutative ring with unity, then the set
of units in R forms a subring. (If true, give a short proof. If false, give
a specic counter-example.)
...
2
votes
1answer
84 views
Prüfer groups are countable
I have read that for any prime number $p$ the Prüfer $p$-group is countable.
My question is: where can I find a proof of this fact?
Thanks.
6
votes
3answers
77 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 ...
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) = ...
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) ...
5
votes
0answers
46 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 ...
2
votes
1answer
46 views
Zero divisors in crossed group rings
It is not difficult to see that a group ring $K[G]$ ($K$ a domain) has non-trivial zero-divisors whenever there exists a non-trivial torsion element $g\in G$. [In fact, in this case, $1-g$ is such a ...
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 ...
2
votes
1answer
46 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 ...
1
vote
0answers
45 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 ...
25
votes
3answers
312 views
Alternative proofs that $A_5$ is simple
What different ways are there to prove that the group $A_5$ is simple?
I've collected these so far:
By directly working with the cycles: page 483 of ...
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 ...
2
votes
2answers
205 views
Cayley tables for two non-isomorphic groups of order 4.
Not sure how to make tables, but:
For a binary operation $*$, and set $\{e,a,b,c\}$, in the Cayley table, $a*a$ can be filled with either the identity or an element different from both $e$ and $a$. ...
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
...
0
votes
2answers
32 views
how do I calculate the ismorphism group of a six-nodes-tree?
How do I calculate the ismorphism group of a connected six-nodes-tree? The tree has a node centred and the other 5 nodes are leaves of the graph. I already know the answer is 6, which is the quotient ...
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 ...
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 ...
2
votes
2answers
57 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 ...
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 ...
6
votes
1answer
44 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 ...
1
vote
2answers
89 views
Is there a residually finite group not finitely presented?
I am looking for a residually finite group which is not finitely presented. Does such a group exist?
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 ...
-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 ...
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 ...
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
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 ...
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 ...
5
votes
2answers
535 views
Why learn Category Theory in order to study Group Theory?
I am self-studying Hungerford's book Algebra. He uses a whole section to talk about categories. In the next section (Direct Products and Direct Sums) he proves that the category of groups has a ...
3
votes
1answer
64 views
Sylow $p$-Subgroups of Classical Groups over $\mathbb{F}_p$
Let $p$ be a prime, and let $G$ be any of the finite classical groups $SL_n(\mathbb{F}_p)$, $O_n(\mathbb{F}_p)$, or $SP_n(\mathbb{F}_p)$. Let $P$ be a Sylow $p$-subgroup of $G$. What is $P$ as a ...
4
votes
1answer
57 views
Sylow $p$-subgroups of $SO_3(\mathbb{F}_p)$
Let $G=SO_3(\mathbb{F}_p)$. We have $|G|=p(p^2-1)$. Let $n_p$ be the number of Sylow $p$-subgroups of $G$. Is it true that $n_p=p+1$?
We have the two conditions
$n_p\equiv 1\mod p$
$n_p\mid ...
4
votes
2answers
124 views
Definitions in a Theorem of Lang
I'm trying to understand the following theorem due to Serge Lang (Algebraic Groups over Finite Fields, 1956, Theorem 2):
Let $p$ be a prime, and let $k$ be a finite field of $q=p^n$ elements. Let ...
4
votes
2answers
87 views
Definition of Unipotent in Positive Characteristic
Let $G$ be an affine algebraic group over an algebraically closed field $k$ whose characteristic is $p>0$. Can $\mathcal{U}(G)$, the set of unipotent elements of $G$, be characterized as all ...
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 ...
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 ...
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 ...
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 ...
0
votes
3answers
190 views
Showing the group of integers modulo m with multiplication is a group
I've just started group theory and I don't know how to show this. Are you supposed to take examples of two elements from the group and an example modulo m (say 2 and 5, and modulo 3) and show the ...
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?
2
votes
0answers
22 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 ...
