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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Cyclic groups and other groups

Ok So I know that a cyclic group is a group that is generated by a single element like $\large{(Z_n,+)}$. Now I was wondering that if every group has a generator , and I found that the answer is yes ...
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55 views

Prove $Inn(G)$ is a normal subgroup of $Aut(G)$ [closed]

I found documents explaining how to prove $Inn(G)$ is a normal subgroup of $Aut(G)$, but can we prove for $G$? Also, I have another question along the same line. How do we show $Aut(G)/Inn(G) \cong ...
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Poker Chips function [closed]

Suppose a poker player has a stack of 33 green poker chips and another stack of 33 white poker chips he then sticks these two stacks next to each other and then mixes the stacks together in an ...
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1answer
59 views

How to show $1+\sqrt 2$ generate an infinite cyclic group of units in $\mathbb Z[\sqrt 2]$?

The answers given here seem very convoluted: The units of $\mathbb Z[\sqrt{2}]$. Is it possible to provide a more explanatory proof?
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1answer
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What does the category of $G$-set look like when $G = C_p$?

Let $G$ be a finite group. The category of $G$-set consists of finite $G$-sets as objects and $G$-equivariant maps as morphisms. Each finite $G$-set is isomorphic to a disjoint union of $G/H$'s, where ...
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$\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$ possible?

Is it possible to have $\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$? My question comes from the link beetween covering and field extensions. For covering the simplest example is ...
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32 views

Arrangements of sets of k positions in a n-competitors race

Let $E(n)$ be the set of all possible ending arrangements of a race of $n$ competitors. Obviously, because it's a race, each one of the $n$ competitors wants to win. Hence, the order of the ...
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40 views

infinite order of element with element in an infinite group

If $G$ is a infinite group, then $G$ must have an element of infinite order. Is this true? I know that if $G$ is infinite cyclic, then it's isomorphic to $\mathbb Z$. (I guess fact is irrelevant ...
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50 views

Order of all elements $(\mathbb{Z}/7161\mathbb{Z}^*)$ divisor of 30

How do I show that for all $x \in (\mathbb{Z}/7161\mathbb{Z})^*$ the order of $x$ is a divisor of $30$? I thought about using the Chinese Rest Theorem, but I don't know how. Alsois there an element ...
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49 views

Must a non-simple group have a normal Sylow subgroup?

In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup. I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow ...
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Locally graded group with all proper subgroups abelian

A group $G$ is said to be locally graded if every finitely generated nontrivial subgroup of $G$ contains a proper subgroup of finite index. I have to prove that a locally graded group with all proper ...
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Group Properties - “$a$” commutes “$b$”?

Dr. Pinter's "A Book of Abstract Algebra" presents this problem from Chapter 4: If $a$ and $b$ are in $G$ and $ab=ba$, we say that $a$ and $b$ commute. Assuming that $a$ and $b$ commute, prove the ...
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I need URGENT help!!! For my Matura Tests Big questions! [closed]

at June I have a test called matura. It is very hard test that it contains subjects from university content, it is a test that every high school student do. I barely understand those subjects, I will ...
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4answers
48 views

How do I show that $b^8=a^3ba^{-3}$?

Suppose $G$ is a group. I am trying to show that for $a,b\in G$, if $aba^{-1}=b^2$, then $b^8=a^3ba^{-3}$. I am not even sure if this is true but I found this in Artin's Algebra. My work: ...
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22 views

substraction of groups in direct sum

Assume I have a sub group $G\leq \mathbb Z^n$ and I have $\mathbb Z^n = G\oplus \mathbb Z^m $ for some $m\leq n$. I want to deduce $G\cong\mathbb Z^{n-m}$. Is that true? how can I do it?
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13 views

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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30 views

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
21 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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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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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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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
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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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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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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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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
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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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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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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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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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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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1answer
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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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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
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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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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, ...
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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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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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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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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 ...