# Tagged Questions

71 views

### Do those 'groups' have a name?

Is there a name for 'groups' that only have a neutral element on the right and an inverse for each element on the right ? If there is, does that name also hold for a neutral elt on the right and an ...
2k views

### What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
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### Forms - unitary group?

If $Id$ is the $n$ by $n$ identity matrix and $J$ is the $2n$ by $2n$ matrix with $Id$ in the upper right corner and $-Id$ in the lower left corner, then $Sp_{2n} = \{G\in Gl_{2n} : G^{tr}JG = J\}$ is ...
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### On the definition of commutators

We know that given a group $G$, for every $x,y\in G$ we define $[x,y]:=x^{-1}y^{-1}xy$, the commutator of $x$ and $y$. I saw something more general, commutators involving more than two elements, like ...
26 views

### What is the Support of a permuation

Is this definition of the support of a permutation correct: let $\pi\in S_{\Omega}$ for $\Omega$ a finite set, and $S_\Omega$ the set of all permutations (bijections) on $\Omega$. Ie ...
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### Solvable Group, which Quotients need to be Abelian?

In Wikipedia it says a group $G$is solvable if it has a subnormal series $\{e\}=G_1\lhd G_2,\dots \lhd G_n=G$ where $G_i$ is a normal subgroup of $G_{i+1}$ and all the factor groups are abelian. My ...
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### Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
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### If a “group” has two identities then is not a group

The story goes like this: A friend and I found this old exercise: Let $G=\Bbb R-\{-1\}$ and $a*b:=a+b+ab$, is $(G,*)$ a group? I say that $(G,*)$ is not a group because for any $a\in G$ follows ...
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### degree of commutativity

What is the exact definition of the degree of commutativity of a $p$-group? When we use notations $d(G)$ and $c(G)$ for other concepts, what is the best notation for degree of commutativity of $G$?
330 views

### What is a Tensor Product?

If you were to explain the concept of a tensor product to an undergraduate(post linear algebra), how would you do so? I would like to hear your definition, your take, on the definition of a tensor ...
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### Definition of a group

What defines a group mathematically, please explain both in Mathematical language and in English if possible. My current understanding: Four things are required to define a group: Closure - Any ...
68 views

### Generator of all congruence classes

Is it possible for $\langle a \rangle =\mathbb Z/n\mathbb Z^*$ for $a\in \mathbb Z/n\mathbb Z^*$? I think I recall hearing it is, what is the name this element takes? I remember hearing generator ...
60 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..
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### Definition question of convex orbit of finite group action

Assume that a finite group or discrete group $G$ acts on a manifold $M$. Here what does it mean that orbit $G\cdot x$ is convex ? Thank you in advance.
105 views

### Understanding equivalent definitions of left cosets

I understand the standard definition of left coset, but the one I do not understand (or see why it is advantageous) is the definition that follows: Let $H\leq G$. Then a left coset of $H$ is a ...
183 views

### What's the difference between Abstract Algebra and Group Theory?

I'm slowly beginning a student of certain higher mathematics. I'm trying to see if I would prefer to study Group Theory or Abstract Algebra. I know that Abstract Algebra seems to "come before" ...
125 views

### Normal subgroup if conjugate subgroup is subset

I find this explanation in Isaacs' Algebra: Lemma. Let $H\subseteq G$ be a subgroup. Then $H$ is a normal subgroup if $H^g\subseteq H$ for all $g\in G$. The reader should be warned that this ...
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### $P \cap N$ 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 ...
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### The relationship between inner automorphisms, commutativity, normality, and conjugacy.

An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$ I have three somewhat broad questions about this: Why is it related to ...
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### Definition of multiple sum

Suppose we have an abelian group $(G,+)$. What is the formal definition of multiple sums such as $\sum_{i_1 \in A} \sum_{i_2 \in A_{i_1}} \cdots \sum_{i_n \in A_{i_{n-1}}}f(i_1,\ldots,i_n)$? Thanks ...
186 views

### Is this an equivalent definition of a normal subgroup?

Let $G$ be a group and $N$ a subgroup. Consider the condition $$(\forall g\in G)(\exists x,y\in G)\ NgN=xNy.\tag1$$ If $N\lhd G$, then for each $g\in G$ we have $NgN=gNN=gN=gN\cdot1$, so the ...
269 views

### What are central automorphisms used for?

A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$. It's not difficult to prove that the set of central automorphisms forms a subgroup of ...
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### 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 ...
168 views

### Definition of subgroup generated by a subset

I'm confused about the definition of a subgroup $(W)$ generated by a subset $W$ of a group $G$. My textbook gives: Let $(W)$ be the set of all elements of $G$ representable as a product of ...
301 views

### Is the free group on an empty set defined?

I'm guessing that the free group on an empty set is either the trivial group or isn't defined. Some clarification would be appreciated.
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### Formula for Product of Subgroups of $\mathbb Z$, Problem

What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$? Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, ...
695 views

### What does it mean to have no proper non-trivial subgroup

I am reading a first course in abstract algebra and there is a claim that says a group $G$ with no proper non trivial subgroups is cyclic. But I don't understand what does it mean to have no proper ...
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### Do dihedral groups $D_n$ for $n\geq 5$ exist?

I know we can generate dihedral group of order three ($D_3$) and four ($D_4$) but my question is whether we can generate dihedral group of order five?
689 views

### Congruent Modulo $n$: definition

In an Introduction to Abstract Algebra by Thomas Whitelaw, he gives examples of the congruence mod operation, such as $13 \equiv5 \pmod4$, and $9 \equiv -1 \pmod 5$. But when I first learned about ...
107 views

### Standard definition of group isomorphism

ProofWiki defines a group isomorphism as a bijective homomorphism. In Topics in Algebra 2$\varepsilon$, Herstein defines a group isomorphism as an injective homomorphism: Definition. A ...
131 views

### What is a minimal polynomial of a group element, and why would we care if it was quadratic?

EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question. I am trying to understand the definition of a p-stable group. The ...
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### Use of the term “normal section” in a theorem of Maria Lucido.

Prop. 3 in this paper (p.135) states Let $G$ be a solvable group with $\text{diam}\Gamma(G)=4$. Then either $l_F(G)\leq 3$ or $l_F(G)=4$ and $G$ has a normal section isomorphic to $H$. ($H$ is ...
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### Is $\{-2,2\}$ a group under $a\star b=\max\{a,b\}$?

Lets say $G={-2,2}$ and $a*b=\text{max}\{a,b\}$. I need to check if this is a group and if it does than is it abelian or not and finite or not. Well... first, I'm not sure if this is a group. for ...
210 views

### Is there an idea of “primeless isomorphism” studied somewhere in finite group theory?

What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't ...
63 views

### What is the “spectrum of $L^1(G)$”?

If $G$ is a locally compact abelian group, what does "the spectrum of $L^1(G)$ mean?" This comes from Folland's A Course in Abstract Harmonic Analysis. As I understand it, $L^1(G)$ is the integrable ...
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### What is the name of the group linear functions on a finite field?

More precisely what is the name for the group $$\{ X\mapsto \alpha^2X+\beta : \alpha,\beta \in GF(q), \alpha \neq 0\}$$ I've always called it the special affine group, but I see that can mean ...
627 views

### Why is associativity required for groups?

Why is associativity required for groups? I'm doing a linear algebra paper and we're focusing on groups at the moment, specifically proving whether something is or is not a group. There are four ...
69 views

### Concerning the point stabilizing group and coset stabilizing group.

I would like to know more about the point stabilizer group and the coset stabilizer group, like the definitions, why they are used in group theory, who developed them and there importance.
132 views

### Is the “binary operation” in the definition of a group always deterministic?

The introduction to group theory that I'm reading requires that the actions of a group are "deterministic"; but the formal definition given makes no mention of this property: A set G is a group if ...
625 views

### Free group and universal property

I'm trying to understand universal properties. An example is the definition of a free group (as I understand it so far): Revised definition: A free group $F_S$ over a set $S$ is a pair $(g,F_S)$ ...
242 views

### Solvability and Simplicity

I have read that Burnside's theorem implies that a group with order $p^aq^b$ cannot be simple. So I looked up Burnside's theorem and saw that it doesn't mention "simple" explicitly, rather it says ...
351 views

### Why is closure omitted in some group definitions?

In some texts, there are three group axioms and in some there are four. The difference is that one of the axioms, the closure ($a,b\in G$ then $a*b \in G$) is omitted. Why is this so?
Let $A$ and $B$ be two subgroups of the same group $G$. What does it mean for the subgroup $A$ to be normalized by the subgroup $B$?
I'm having trouble understanding the way the identity element is defined in Lang's Algebra. Below is the relevant information. Suppose we have a monoid G with elements $x_{1},...,x_{n}$. We can define ...