Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.
0
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17 views
Why don't we consider non-units as quadratic residues?
Is there any specific reason in not including non-units of $\mathbb{Z}_n$ as quadratic residues? As an examples, we say that in $\mathbb{Z}_8$, the set of quadratic residues is just {1} and not {1,4}.
...
1
vote
2answers
29 views
Give an example of the $a,b,c$ which satisfies conditions in the generating set
How to derive the specific case of the generating element of a group given its generating set. For example, when
$$G=\langle a,b,c|a^3=b^3=c^2=1,ab=ba,ca=a^2c,cb=b^2c\rangle$$
we can let $G\subset ...
1
vote
0answers
13 views
Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)
This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups.
More precisely, given discrete groups below (a), (b), (c):
...
4
votes
3answers
82 views
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at
least two elements.
We know that Aut($G$) contains the identity function $f: G \to G: x \mapsto x$.
If $G$ is ...
3
votes
1answer
101 views
Question on groups of order $pq$
Let $G$ be a group of order $pq$, $p>q$ and $p$, $q$ are primes. Then prove that
If $q\mid p-1$ then there exists a non abelian group of order $pq$.
Any two non-abelian groups of ...
4
votes
1answer
86 views
On Group of order $30$ and $60$.
In this question on yahoo answers ,
the answer says ,
"with $t = 6$, then there are 6 * (5 - 1) = 24 elements of order $5$ "
my question is , how did " 6 * ( 5 - 1 ) " come from ?
Which ...
1
vote
1answer
26 views
Symmetry of Plancherel measure (for $S_n$)
For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$:
$$
\begin{pmatrix}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 ...
2
votes
1answer
52 views
About commutators and center o a certain group
Let $G$ be a group and $H, K\unlhd\ G$ (normal subgroups), with $K\leq H$ such that $\left[H, Aut(G)\right]\leq K$. Why does this imply that $H/K\leq Z(G/K)$? $Z(G)$ denotes the center of $G$.
3
votes
0answers
52 views
Discrete subgroups of SU(n) and SO(n).
Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful.
I would like to know whether there is a complete understanding of discrete ...
3
votes
1answer
45 views
Action of $S_7$ on the set of $3$-subsets of $\Omega$
Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem:
Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
7
votes
3answers
240 views
Presentation of Rubik's Cube group
The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
4
votes
3answers
225 views
Show a certain group is contained in a Sylow p-group.
Statement: Let G be a group and p a prime that divides $|G|$. Prove that if $K\le G$ such that $|K|$ is a power of p, K is contained in at least one Sylow p-group.
I just started studying Sylow ...
2
votes
2answers
31 views
Finite group is generated by a set of representatives of conjugacy classes.
Could you tell me how to prove that a finite group is generated by a set of representatives of conjugacy classes?
I've read this ...
1
vote
2answers
94 views
Show groups of symmetries of a cube and a tetrahedron are not conjugate in isometry group.
I've shown that the symmetry group of a cube and a tetrahedron are both isomorphic to S4, but I am now trying to show that they are not conjugate when considered as subgroups of isometries of 3D ...
3
votes
0answers
71 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 ...
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 ...
4
votes
1answer
45 views
Group homomorphisms into a field
Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
3
votes
1answer
61 views
Infinite products of a (finite) group
So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
0
votes
2answers
56 views
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$.
Here is what I have here:
Aut$(G)$ consists of 6 bijective functions, which maps $G$ to ...
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 ...
45
votes
0answers
848 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 ...
4
votes
2answers
94 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.
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 ...
25
votes
3answers
315 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
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 ...
-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)$.
7
votes
2answers
138 views
Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.
What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve ...
3
votes
3answers
111 views
Sylow $p$-subgroups of Finite Matrix Groups
Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$?
I believe the answer is yes, because I ...
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 ...
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.
5
votes
3answers
99 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?
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 ...
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 ...
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
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 $x \in X$ ...
0
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1answer
61 views
4
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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
1answer
54 views
Abelian groups related to $(\mathbb{Z}/p\mathbb{Z})^*$, with order close to $p$ — analogous to elliptic curve groups, but simpler
I'm looking for a construction of a group $H$ that is "a sister" to $(\mathbb{Z}/p\mathbb{Z})^*$, in the following rough sense:
Each element of $H$ can be represented by one or a few elements 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, ...
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 ...
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
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 $p=5$)?
0
votes
0answers
46 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
65 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 ...
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 ...
13
votes
1answer
196 views
Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$
I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following:
Let $G$ be a finite group ...
2
votes
1answer
61 views
The indecomposable projective $\mathbb{F}_pG$-module with $UJ/J\cong \mathbb{F}_p$
Let:
$G$ be a finite group;
$p$ be prime;
$J$ be the Jacobson radical of $\mathbb{F}_pG$.
A paper I'm trying to read mentions the following object:
The indecomposable projective ...
