A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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The concept of continuity in a topological group

I am now learning the Lie group theory. People talk about the fundamental group of a topological group. The problem is, how is the continuity defined in a topological group? In other words, in which ...
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Test for a $G$-torsor to be trivial?

I just have a very short question, why is a $G$-torsor trivial precisely when it has a section?
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1answer
64 views

Is there a cyclic subgroup C of S8 such that the interval lattice [C,S8] is distributive?

I've checked by hand that for any $n \le 7$, there is a cyclic subgroup $C$ of $S_n$ such that the intermediate subgroups lattice $\mathcal{L}(C \subset S_n)$ is distributive. Question: Is it the ...
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40 views

normal subgroup in $S_3$?

Is $\{(1),(1,3)\}$ a normal subgroup in $S_3$? I know that a normal subgroup means that the left cosets are equal to the right cosets.
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1answer
22 views

Short exact sequence of abelian groups implies long exact sequnce of cohomologies

I am trying to compute cohomologies $H^i(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}, \mathbb{Z})$. Actually it is not a big deal, because I have already computed $H^i(\mathbb{Z}/n\mathbb{Z}, ...
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Prove that $A$ is a free abelian group

Suppose $a_1, \dots, a_n$ generate an abelian group $A$, and for any abelian group $B$, and any $b_1, \dots, b_n \in B$ we can find a homomorphism $\varphi: A \to B$ given by $\varphi(a_i) = b_i ~ ...
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1answer
35 views

Right group axioms from left group axioms

I was working a question in group theory where we are given that the left axioms hold for a set $G$ together with a binary operation $*$. We would then like to prove that $(G, *)$ is a group. I read a ...
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80 views

Group of Order $5$

Let $G$ be a group of order $5$ with elements $a, b, c, d, 1$ where $1$ is the identity element. This is the definition of the group. We all know that this can't be a group because any group of ...
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35 views

Consequences of $G$ having exactly $3$ irreducible characters

Suppose a finite group $G$ has exactly irreducible characters $\chi_1 = \mathbb{I}, \chi_2,\chi_3$. i) Show that G is soluble and deduce it has a non-trivial one-dimensional character $\chi_2$. ii) ...
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Semi-linear transformations form a group.

Let $K$ be a field and $a,b,c,d\in K$ such that $ad-bc\neq 0$ and $\sigma \in Aut(K)$, we define a semilinear transformation $f: K \cup \lbrace \infty \rbrace \to K \cup \lbrace \infty \rbrace$ ...
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1answer
32 views

Proof that elements of a free generating set have infinite order.

I'm trying to show that elements of a free generating set $S$ have infinite order straight from the definition of a free group being generated by $S$. The definition I'm using is that a group $F$ is ...
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139 views

Why isn't $\mathbb{Z}_2 \times \mathbb{Z}_{30}$ isomorphic to $\mathbb{Z}_{60}$?

I know that other groups like $\mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_5$ are isomorphic to $\mathbb{Z}_{60}$.
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1answer
46 views

Does all splitting fields have characteristic 0?

Does all splitting field have characteristic 0? This may be a bad question, but I am wondering because the author of a book I am reading summarises a lot of properties when he is about to start with ...
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1answer
67 views

Is such set a group?

Question: If a set $G$ is equipped with an associative binary operation $\ast$, and assume $G$ has identity element $e$ and for each $g \in G$ there exists its inverse element $g^{-1}$, is $G$ a ...
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1answer
25 views

Is the commutator subgroup functor exact?

I'm wondering whether the commutator subgroup functor on the category of groups is an exact functor in the sense that it preserves exact sequences of the form $A \rightrightarrows B \to C$ where the ...
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2answers
38 views

A subgroup of $\textrm{GL}(3,q)$ of order $q^2(q-1)$

Let $q$ be a prime power. Consider the multiplicative group $\textrm{GL}(3,q)$ of the $3 \times 3$ matrixes with coefficients in $\mathbb{F}_q$ which are invertible. The matrixes $$ M_{a,b,c} = \left( ...
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0answers
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Generalization of Cauchy-Davenport inequality to $\mathbb Z_p^d$

Is there some generalization of Cauchy-Davenport inequality to the group $\mathbb Z_p^d$? ($p$ prime number, $d \ge 3$) For example, Kneser Theorem says that if $G$ is any abelian group and ...
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1answer
45 views

$S_4$ is not supersolvable? Why am I wrong?

I read that $S_4$ is an example of a solvable group who is not a supersolvable group. In order to prove it is solvable, we see that: $\{e\}<\{(1),(12)(34)\}<K<A_4<S_4$ where $K$ is the ...
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1answer
40 views

What is a stabilizer of a matrix (group action is multiplication)

I have the group $GL(3,\mathbb R)$ (the group of invertible 3x3 matrices) acting on $M_{33}(\mathbb R)$ (the set of 3x3 real matrices) by $A\cdot M=AM$. Let $$M_1 = \begin{pmatrix} 1 & 0 & 1 ...
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1answer
33 views

Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field?

Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field? I'm confused because the polynomial $x^2+1$ in $F_2[x]$ is inseparable ($f(x)$ and ...
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34 views

Determine Isomorphism type

Determine isomorphism type of quotient group $$\mathbb{Z} \times \mathbb{Z} / \langle(1,1)\rangle $$ using Fundamental Theorem Finite Generated Abelian Groups after looking at the factor group, it ...
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38 views

Group Theory-Isomorphisms

Currently in Abstract Algebra, discussing group theory. In order to show two groups are isomorphic to each other, I know what you need to show, $1$-$1$, onto, and homomorphism. what I'm having a ...
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3answers
41 views

Is there any permutation $\tau\in S_7$ so that: $\tau^{4}=\sigma$?

Let $\tau$ be a permutation in $S_7$: $\ \sigma= \left( \begin{matrix} 1 & 2 & 3 & 4&5&6&7\\ 3 & 4 & 5 &6&1&7&2 \ \end{matrix} ...
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1answer
36 views

Product of a character with an irreducible character a non-negative integer

Why is the inner product of a character with an irreducible character a non-negative integer? I can see that by properties of the inner product it will be non-negative but I cannot see why it would ...
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48 views

There is an element, whose order is the exponent of $H$

If $H$ is a subgroup of $K^*$, where $K$ is an arbitrary field, then there is an element $h\in H$, whose order is the exponent of $H$, that is the least common multiple of the elements of $H$ I ...
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243 views

Is finiteness necessary in this exercise?

This is from Dummit&Foote Abstract Algebra Chap 3.1 Problem 28. Here is problem & solution of this problem. Let $N$ be finite subgroup of $G$ and suppose $G=\langle T\rangle$ and $N=\langle ...
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140 views

Maximum number of Sylow subgroups

I've been studying Sylow-$p$ subgroups, and I've come across this problem. Let $G$ be a finite group. Show that the number of Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$ . ($|G|$ is the ...
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1answer
76 views

Proof that $\gcd(2^m-1,2^n+1)=1$ for odd $m$ using group theory

Below is a perfectly fine proof using basic tools of number theory: Showing $\gcd(2^m-1,2^n+1)=1$ Could we prove this more quickly using group theory? I would be very interested in seeing an ...
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1answer
19 views

Group Determinant Independent of Labeling of Elements

Let $G$ be a finite group with elements $g_1, g_2, \ldots, g_n$. We define the group matrix by $$X_G = [x_{g_ig_j^{-1}}].$$ We then can define the group determinant as $$\det X_G = \Theta_G.$$ ...
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59 views

if o(a) is equal to exponent of finite abelian group G then $G=<a>\times K$

problem:prove that if $o(a)$ is equal to the exponent of a finite Abelian group $G$, then there exists $H<G$ such that $G=H\times\langle a\rangle$$$$$ using fundamental theorem of finitely ...
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48 views

Quotienting by generators in free groups

I feel like this is a simple result but have not touched algebra in a while and can't find the right combination of words to search for. Suppose we have a free group on 2 generators $G = \langle a, ...
3
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1answer
38 views

Subset being an Abelian group.

There is a theorem that states: Let $P(X)$ be the set of subsets of $X$ and let $$\Delta $$ be the symmetric difference defined by $$A\Delta B = (A \cup B)\text{$\backslash $(A $\cap $ B)}.$$ Then ...
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1answer
48 views

Multiplication modulo 10 in Cayley's Table

In the Fourth row and second column, we have $3\cdot7 = 21$ But $9$ is the "limit", and note that there are $4$ elements in $U(10)$ Using modular arithmetic, $21= (4*5)+1$ Thus, $21 = 1 \pmod 4$ ...
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45 views

What automorphisms exist on the abelian group of positive rationals under multiplication?

Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$. If we relax the problem to ...
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1answer
96 views

Eigenvalues of Group Elements and Quaternions

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
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1answer
29 views

Question about Eigenvalues of group elements

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
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42 views

Isomorphisms and integers

One more question for me. Is it incorrect to say that (m$Z$,+) is isomorphic to (n$Z$,+) because both are infinite cyclic groups that are isomorphic Z under addition?
2
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1answer
30 views

Homomorphisms Groups Kernel

Given G: group of units in Z mod 14 under multiplication. A function sends the integers under addition to G. $f(n)$ = $[3]^n$ I am just checking whether I am correct in stating that the kernel ...
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3answers
40 views

'Inverse' property of a group and the special case that makes a group an Abelian group

One of the property for which a set must have in order to be a group is to possesses the 'inverse' property. What this says is that for each element $a$ in $G$, there is an element $b$ in $G$ with ...
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1answer
49 views

Can a group be non-empty by definition of 'group'?

By the definition of a group, "A group is a set combined with a binary operation". By this definition, would a non-empty set constitute as being a group? By virtue of the definition of what a group ...
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0answers
28 views

Prove or provide counterexample that this defines a homomorphism of groups.

Let $X=\langle x \rangle$ be a cyclic group generated by $x$. Let $G$ be a group and $\langle h \rangle$ a subgroup of $G$. Suppose the order of $h$ divides the order of $x$. Prove or provide a ...
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Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
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The set of all the rotations of a plane around multiple points contained in it doesn't form a group.

I'm reading the book "An introduction to the theory of Groups" by P.S. Alexandroff and in one of his examples he says that the group of all the rotations of a plane around a given (fixed) point forms ...
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70 views

Need help determining what this center is isomorphic to

I am looking at this released exam, problem 2c. It states: Let $G$ be a non-abelian group of order $8$ and $Z$ be the center of $G$. To which group is $Z$ isomorphic? It gives a hint to recall a ...
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43 views

A finite group which is isomorphic to $PSL(2,p)$

Let $p$, $q$ be a odd prime numbers such that $p=2q+1$. Let G be a finite group of order $\frac{p(p^2-1)}{2}$. If all Sylow 2-, p-, and q-subgroups are not normal, G is isomorphic to $PSL(2,p)$. The ...
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1answer
41 views

Is O(1) a Lie Group?

In reading Georgi (Lie algebra in particle physics) I reaf at page 43 the following definition of Lie Gruoup: "a lie gruoup is a group whose elements depend smoothly on a set of continuous ...
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unitary transformation of $\mathbb{C^2} \otimes \mathbb{C^2}$ that preserves the decomposability

I have some doubts about the relationship between tensor product and unitary transformations... Take $\mathbb{C^2} \otimes \mathbb{C^2}$ and think about it as a inner product space with the canonical ...
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How to construct generators and Lie Algebra for Lorentz group?

I'm trying to figure out Lorentz group in 2+1. First of all, I am physicist and I'd like to think the special orthgonal group as a combination of rotation and translation in space. Then I construct it ...
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2answers
244 views

Every subgroup of a normal subgroup is normal

Is every subgroup of a normal subgroup normal ? That is if $H$ is a normal subgroup of a group $G$ and $K$ is a subgroup of $H$, then $K$ is a normal subgroup of $G$. Is it true ? If not what is the ...
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Action of $SL_2(p)$ on intgers mod p by Möbius transformations

This is simple and may have been asked before, but I couldn't find it. I have been asked to 'Define the action of $SL_2(p)$ (the group of 2 by 2 matrices of determinant 1 with entries in ...