2
votes
1answer
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 ...
23
votes
6answers
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 ...
0
votes
1answer
10 views

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 ...
0
votes
1answer
37 views

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 ...
1
vote
1answer
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 ...
1
vote
0answers
36 views

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 ...
1
vote
0answers
39 views

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 $G$ ...
3
votes
1answer
71 views

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 ...
3
votes
2answers
64 views

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 $G$?
0
votes
2answers
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 ...
1
vote
1answer
81 views

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 ...
1
vote
1answer
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 ...
6
votes
1answer
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..
0
votes
1answer
59 views

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.
2
votes
1answer
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 ...
2
votes
1answer
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" ...
1
vote
2answers
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 ...
1
vote
1answer
151 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 ...
3
votes
1answer
170 views

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 ...
0
votes
1answer
53 views

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 ...
14
votes
2answers
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 ...
6
votes
2answers
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 ...
5
votes
2answers
149 views

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 ...
0
votes
4answers
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 ...
9
votes
4answers
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.
1
vote
3answers
144 views

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, ...
1
vote
2answers
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 ...
-3
votes
3answers
132 views

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?
7
votes
4answers
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 ...
4
votes
2answers
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 ...
4
votes
4answers
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 ...
1
vote
1answer
49 views

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 ...
5
votes
5answers
171 views

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 ...
7
votes
2answers
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 ...
0
votes
1answer
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 ...
2
votes
1answer
82 views

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 ...
6
votes
5answers
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 ...
0
votes
0answers
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.
2
votes
1answer
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 ...
3
votes
1answer
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 $(g,F_S)$ ...
1
vote
2answers
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 ...
8
votes
3answers
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?
0
votes
1answer
555 views

normalized subgroup by another subgroup

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 $B$?
1
vote
2answers
111 views

Definition of identity in a monoid

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 ...