The study of symmetry: groups, subgroups, homomorphisms, group actions.
1
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0answers
4 views
Left regular action isomorphic to the right regular action
I am reading something on regular groups and I have a question on why the left and right regular actions are isomorphic.
Let $G$ be a group. Consider the homomorphism $\rho: G \rightarrow Sym(G), g ...
2
votes
1answer
18 views
The structure of $(\mathbb Z/525\mathbb Z)^\times$
I am working on the following problem.
Find the number of the elements of order 4 in $(\mathbb Z/525\mathbb Z)^\times$.
I tried to solve it in the following way: since $525=3\cdot5^2\cdot7$, we ...
1
vote
1answer
44 views
Group presentations and subgroups
How to prove that $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ has no subgroup of order $6$ without finding $G$?
2
votes
1answer
40 views
Subgroup complement for normal subgroup of $G$ with trivial center and $\mathrm{Aut}(N)=\mathrm{Inn}(N)$
Let $G$ be a finite group and $N \subseteq G$ be a normal subgroup. If $Z(N)$ is trivial and $\operatorname{Aut} N=\operatorname{Inn} N$, then show $N$ has a complement $H$ and $H$ is normal in $G$ ...
3
votes
4answers
66 views
A presentation of a group of order 12
Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$.
I tried to let $d=ab\Rightarrow G=\langle d,c\mid ...
2
votes
1answer
62 views
Existence of group of order $p$ in group of order $pq$, $p>q$
This question is related to Question on groups of order $pq$, but is different.
It references the same exercise, but an earlier part.
The exercise is: A group of order $pq$, $p>q$, contains a ...
1
vote
1answer
20 views
upper central series, taking intersection with subgroup ($\zeta_i G \cap H \leq \zeta_i H$)
Let $\zeta_i G$ be the upper central series of $G$. Show that for 'any' subgroup $H$, we have
$$\zeta_i G \cap H \leq \zeta_i H$$
where $\zeta_i H$ is the upper central series of $H$.
I tried ...
1
vote
0answers
35 views
finite normal minimal
Suppose that the set of all minimal normal subgroups of $G$ is finite and that each of them is also finite. Let M be the product of them all.
My questions is:
If $C_{G}(M)\neq 1$ is true that ...
0
votes
2answers
35 views
limit of square root of function and example of group
can somebody help in getting the value of
$\displaystyle\lim_{x\to 9}\frac{\sqrt{f(x)}- 3}{\sqrt{x}-3}$, if $f(9) = 9$ and $f(9) = 4$.
Though the problem seems to be very simple, when I tried the ...
1
vote
1answer
33 views
question on group representations
Here is a problem I faced in algebra. $\rho: A_4 \rightarrow End_{\mathbb C}\mathbb C^{10}$ is a representation of $A_4$. Then show that there is a vector $v \in \mathbb C^{10}$ such that $v$ is an ...
2
votes
2answers
54 views
Open Subgroup of (R,+)
Let G be an open subgroup of (R,+)
Show that G=R.
Note: I've tried taking an interior point of G. Can Archimedian Property be used?
1
vote
1answer
38 views
To What Extent Does the Cartesian Product for Algebraic Structures Generalize?
I admit this question is quite general.
If we have a group (or perhaps some other algebraic structure) $G$, we can define the Cartesian product $G\times G$ of $G$ with itself. And then powers of $G$ ...
0
votes
1answer
74 views
Check existence of an isomorphism in four examples
Check if there exists an isomorphism for :
$$ i) \ G = (\mathbb{Z}_{15}, +_{15}), \ H = (\mathbb{Z}_{15}, +_{15}), \ f(1) = 2 $$
$$ ii) \ G = (\mathbb{Z}_{15}, +_{15}), \ H = (\mathbb{Z}_{15}, ...
4
votes
3answers
66 views
metric property in a group
Can we define a metric function on a group $G$? Please give examples other than $\mathbb{R}$. Actually most groups have elements in discrete manner. It sounds vague but I can't be more precise.
3
votes
3answers
71 views
$G$ a group of odd order. Then $\forall$ $g\in G$ there is $h\in G$ such that $g=h^2$
This one is from a practice exam I was working on.
$G$ a group of odd order. Then for $\forall$ $g\in G$ there is a unique $h\in G$ such that $g=h^2$.
Thoughts Well I tried a few things but they ...
0
votes
0answers
30 views
Central quotient of $G=K\rtimes H$ where $H$ is core-free and $Z(G)\leqslant K$
If $G=K\rtimes H$ is a semidiect product of a core-free group $H$ and a normal subgroup $K$ such that $Z(G)\leqslant K$, what can we conclude about the quotient $G/Z(G)$?
1
vote
2answers
58 views
Check condition normal subgroup in these three examples
Is the subgroup H of G is a normal subgroup of G, for:
$$ i)\ G = S_5, \ H = \{id, (1,2)\} $$
$$ ii) \ G = (Sym(\mathbb{N}), \circ), \ H = \{f\in Sym(\mathbb{N}) : f(0) = 0 \}$$
$$ iii) \ G = S_4, \ H ...
3
votes
3answers
46 views
Torsion Subgroup (Just a set) for an abelian (non abelian) group.
Let $G$ be an abelian Group.
Question is to prove that $T(G)=\{g\in G : |g|<\infty \}$ is a subgroup of G.
I tried in following way:
let $g_1,g_2\in T(G)$ say, $|g_1|=n_1$ and $|g_2|=n_2$;
Now, ...
3
votes
2answers
44 views
On the quotient group $\pi_{1}(K)/N$ for the Klein bottle $K$
I know that the Klein bottle $K$ is obtained from the unit square by making identifications on the boundary with the appropriate directional arrows. Usually, what is done is that we identify the point ...
1
vote
2answers
48 views
Let $G$ be a group of order $4p^n$, where $p>2$ is prime and $n>0$. Show that $G$ is not simple.
Let $G$ be a group of order $4p^n$, where $p>2$ is prime and $n>0$. Show that $G$ is not simple. (Hint: Consider the standard action of $G$ on $G/P$, where $P$ is a $p$-sylow subgroup.)
Let ...
1
vote
0answers
54 views
Show that a periodic, finitely generated, nilpotent group has finite order.
Show that a periodic (every element has finite order), finitely generated, nilpotent group has finite order.
4
votes
1answer
60 views
Extending Automorphism of $G$ to $G/N$
In Herstein's Topics in Algebra, there is a problem:
If $G$ is a group, $T$ an automorphism of $G$, and $N$ a normal subgroup of $G$, s.t. $(N)T \subset N$, construct an automorphism of $G/N$.
My ...
2
votes
1answer
58 views
Cyclic (sub)groups
I have two questions that are confusing me.
Why is the cyclic group generated by $g$ the smallest subgroup containing $g$?
If $g$ generates an infinite subgroup, why is it called cyclic? I mean, ...
1
vote
1answer
67 views
Find all homomorphisms in these three examples
How I should find all homomorphisms $$ f : G \rightarrow H $$ for examples:
$$ i) \ \ G = (Z, +), \ \ H = (Q, +) $$
$$ ii) \ \ G = (Z_{15}, +_{15}), \ \ H = (Z_4, +_4) $$
$$ iii) \ \ G = Z_2 \times \ ...
0
votes
4answers
102 views
Prove that if $x^2=e$ then order of $x$ is $1$ or $2$
Let $(G,*)$ be a finite group and $x$ an element of $G$.
Prove that if $x^2=e$ then the order of $x$ can be only $1$ or $2$?
2
votes
1answer
56 views
The maximim number of elements in the Alternating group of degree 28
I know if $r$ is a prime number, then $(r-1)!$ is maximum number of elements of same order in the alternating group $A_{r}$. What is maximum number of elements of same order in the alternating group ...
0
votes
1answer
55 views
To classify finite $p$ group with special property
I wish to classify following finite $p$ group.
Let $G$ be a finite $p$ group with the property whenever $H$ is a non normal subgroup of $G$ of order $p$, $G$ is the semidirect product of $H$ and a ...
-1
votes
0answers
36 views
Which books teaches group theory and generating function
I want to learn how to different group convert to generating function
from permutation group, finite group to generating function
furthermore is q-generating function,
and from view of counting, ...
0
votes
5answers
129 views
Why composition in $S_3$ is not commutative
The family of all the permutations of a set $X$, denoted by $S_X$, is called the
symmetric group on $X$. When $X = \{1, 2, \dots , n\}$, $S_X$ is usually denoted by $S_n$, and it is called the ...
-2
votes
1answer
37 views
set of terminating decimals
Let $T\subset\mathbb Q$ be the set of all positive rational numbers that can be represented by a terminating decimal (in base 10), that is, a decimal whose tail consists of an infinite sequence of ...
3
votes
1answer
32 views
Non vanishing of group cohomology
Let $G$ be a finite group, then $H^n(G,\mathbb{Z})\neq 0$ for infinitely many $n$. This is not hard to see for cyclic groups. Can we prove this fact algebraically, could anyone provide a reference? ...
2
votes
1answer
60 views
A group of order 16 has a normal subgroup of order 4
Let $ G$ be a group of order $16$. Show that $G$ must contain a normal subgroup $H$ of
order $4$.
I tried the Sylow first theorem, that is $\{e\}\triangleleft H_1\triangleleft H_2\triangleleft ...
2
votes
1answer
136 views
Counter examples in group theory
Let $G$ be a finite group with normal subgroups $N_{1}$ and $N_{2}$. Find counter examples to the following statements
1) If $N_{1}\cong N_{2}$ then $G/N_{1}\cong G/N_{2}$
2) If $G/N_{1}\cong ...
1
vote
3answers
38 views
Two questions about isomorphisms of groups
Let $A_1$ and $A_2$ be Abelian groups and let $C$ and $B_i$ be subgroups of $A_i$ for $i=1,2$ and $B_1 \cap B_2 = A_1 \cap A_2 = \{0\}$. Is then true that
$$(A_1\bigoplus A_2) / (B_1\bigoplus B_2) \ ...
1
vote
1answer
88 views
Non-abelian groups where $(ab)^{n+i}=a^{n+i}b^{n+i}$ for all $0\leq i\leq k$, $k>1$
In this question it is asked for an example of a group $G$ such that $(ab)^n=a^nb^n$ for all $a, b\in G$ holds for two consecutive integers $n\in\{m, m+1\}$, but $G$ is not an abelian group. The ...
1
vote
1answer
65 views
Abelian Groups and Number Theory
What is the connection between "Finite Abelian Groups" and "Chinese Remainder Theorem"?
(I have not seen the "abstract theory" behind Chinese Remainder Theorem and also its proof. On the other ...
3
votes
1answer
85 views
If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$
If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$.
I have no idea where to start, but this is my abstract algebra homework, so I ...
1
vote
1answer
24 views
Structure of stabilizer of nonsingular line in $\Omega(7,q)$
Let $q$ be a prime power, $\varepsilon=\pm$, and $M$ be the stabilizer of a nonsingular line in $\Omega(7,q)$ such that $M=\Omega^\varepsilon(6,q).2$. Then can we know more explicit structure of $M$? ...
6
votes
1answer
29 views
Definition of $\Omega$-group and $\Omega$-composition series
What are the definitions of $\Omega$-group and $\Omega$-composition series? No luck searching on the internet..
1
vote
1answer
63 views
A question on subgroups of an infinite permutation group
I have come across this problem in the course of studying for my comprehensive exam in algebra, which is fast approaching. I would appreciate any suggestions and/or hints as to its solution.
Let ...
5
votes
1answer
54 views
$\mathbb{Z^2}/\langle(x, y)\rangle$ as a direct sum of cyclic group
For each $(x,y)\in \mathbb{Z}^2$, let $\langle(x, y)\rangle$denote the subgroup of $\mathbb{Z}^2$ generated by $(x,y)$. Express $\mathbb{Z}^2/\langle(x, y)\rangle$ as a direct sum of cyclic group.
...
0
votes
1answer
26 views
Proving that sum is invariant under bijections.
I am trying to prove the following: let $(G,+)$ be an abelian group, let $I_n^m = \{i \in \mathbb{N} : n \leq i \leq m\}$ and $I_r^k = \{i \in \mathbb{N} : r \leq i \leq k\}$. Let also $\varphi : ...
6
votes
0answers
123 views
Kernels in $\mathbf{Top}$
There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following :
...
0
votes
0answers
53 views
Finite groups and their center
Let $G$ be a finite group. Consider $X = \{[x,y] \mid (\lvert x \rvert, \lvert y \rvert) = 1, x,y \in G \}$ and $K = \left< X \right>$. Suppose that $\vert a^X \vert = m$, for all $a \in G$. If ...
1
vote
0answers
44 views
Equivalence of tensor reps & tensor products of reps
Let a finite-dimensional vector space $V$ over $\mathbb R$ or $\mathbb C$ with dual $V^*$ and a group $G$ be given. Let $\rho:G\to\mathrm{GL}(V)$ be a representation, and let $T_kV$ and $V^{\otimes ...
3
votes
1answer
77 views
A question on a subgroup of an abelian group
Let $X$ be an abelian group, $G$ a divisible subgroup of $X$ and $L$ a subgroup of $X$ such that $L\bigcap G\neq 0$. Let $nX=G$ for some $n$. For a positive integer $m$, define $H_{m}=\{x\in X;mx\in ...
3
votes
0answers
30 views
Supremum over unitary group action
Let $A$ and $B$ are two given Hermitian operators on matrix algebra $M_n(\mathbb{C})$. $A$ is positive semi-definite with unit trace. I want to know the general method for calculating the following ...
1
vote
2answers
88 views
$G$-set terminology
When a group action $G \times X \rightarrow X$ is defined with a group $G$ and a set $X$, why is there not a special name for the set $X$? I know that this is referred to as a $G$-set, but the set $X$ ...
3
votes
2answers
81 views
Two dimensional complex group representations
Michael Artin's Algebra, chapter 10 (both unstarred, and complex representations)
M.8 Prove that a finite simple group that is not of prime order has no nontrivial representation of dimension 2.
...
0
votes
2answers
55 views
Number of distinct groups of order n upto isomorphism, for a fixed integer n.
Given any positive integer $n$, is the number of distinct groups of order $n$ upto isomorphism finite?
Thanks in advance.
