1
vote
2answers
30 views

Difference between the definition of monoid action and group action?

The question is essentially in the title. From what I read in the wikipedia article about monoids it seems to me that we can define a monoid action in the exact same way we define a group action. Is ...
4
votes
0answers
25 views

Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
1
vote
2answers
71 views

Confusion related to definitions involving free groups

From Wikipedia ...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different ...
1
vote
1answer
28 views

Why are isotropy groups named as such?

Why are isotropy groups, also known as stabilizers, named as such? In physics, the word isotropy means having the same property in all directions. Can one draw an analogy from this to interpret the ...
0
votes
1answer
16 views

Terminology question in group action

Given a group action of $G$ on a set $X$, and $S \subset X$, is there a name for the set $H \subset G$ such that for all $g \in H$ and $x \in S$, $g \cdot x \in S$ ? I do not require that $g \cdot x = ...
5
votes
1answer
65 views

The name of a certain type of groups

What is the name (if any) given to groups satisfying: $$\forall x,y,z\in G [xyx^{-1}=(zxz^{-1})y(zxz^{-1})^{-1}]$$ I understand this question might not be suitable here, but I really can't search ...
2
votes
1answer
48 views

How did the Symmetric group and Alternating group come to be named as such?

The Dihedral group makes sense, "Di" means two, and "hedral" means.. shape I think (I've just realised how much of what I think words mean are guesses based on experience) like a "polygon" is a 2d ...
3
votes
1answer
56 views

Why is a perfect group called a perfect group

A group is called perfect if we have $[G,G]=G$. I was wondering in what sense is this group perfect? I've never really done anything much with perfect groups so I don't really know anything about ...
5
votes
2answers
97 views

Is there a name for this object? (Like a group, but the inverse is not necessarily a member of the set)

A group is a set $G$, together with a binary operation $\cdot$ that is closed - if $f\in G$ and $g \in G$ then $f\cdot g \in G$ is associative - $(f \cdot g) \cdot h = f \cdot (g \cdot h)$ has an ...
3
votes
1answer
36 views

Terminology concerning conjugation in groups of functions.

If there is a function $a$ such that $a\circ g\circ a^{-1}=h$ then the functions $g$ and $h$ are conjugate to each other. If one wished to identify $a$, would one say "$g$ and $h$ are conjugate "by ...
2
votes
1answer
43 views

In the transition from set theory to order theory, what is the appropriate generalization of “group”?

Given a set $X$, the collection of all bijective endofunctions on $X$ forms a group, called the symmetric group on $X.$ Furthermore, Cayley's theorems says that every group embeds into a subgroup of ...
2
votes
0answers
50 views

Terminology problem

I am not a mathematician, please forgive my incorrect language. My question involves terminology. If a finite non-abelian group G is represented by a set of unitary operators ${\mathbf G}_r, r = ...
0
votes
1answer
36 views

Notation for permutation corresponding to the action of a group element

Let $G \times X \to X,\ \ (g,x) \mapsto g.x$ be an action of $G$ on $X$, i.e., $e.x = x$ for all $x \in X$; $gh.x = g.(h.x)$ for all $g \in G$, $x \in X$. For a fixed $g \in G$, how should I refer ...
1
vote
0answers
38 views

First Homomorphism Theorem and terminology

I am not finding clear terminology in my abstract algebra book to be clear at least and my questions are simple. Consider the construction of a quotient group G': \begin{equation} G/K = G' ...
0
votes
0answers
61 views

Terminology on group actions

Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. My knowledge of group theory is undergraduate-level stuff. I'm looking at the paper cited ...
1
vote
3answers
57 views

What does “nilpotent” in a “nilpotent group” mean?

It seems to have nothing to do with the usual nilpotency, i.e. $\exists n\in \mathbb{N}:x^n=0$. Actually I think the latter only makes sense in a ring or more rich structure. I tried to relate some ...
5
votes
0answers
121 views

Why are they called orbits?

When we study actions in group theory, we consider sets of the form $$\text{Orb}_G(x)=\{gx\mid g\in G\} $$ that are called orbits. Although, the only reason I find convincing for that name is that in ...
4
votes
1answer
75 views

On group theory terminology

Let $G$ be a finite group. Consider the next number $$m(G):=\min\{m\in\mathbb{N}\mid G\ \text{can be embedded into}\ S_{m}\}.$$ It is obvious that Cayley's theorem yields $m(G)\leq |G|$. My ...
6
votes
1answer
59 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
2answers
103 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 $X$ ...
1
vote
1answer
79 views

Why are regular $p$-groups called “regular?”

In the concept of regular $p$-Groups, what does "regularity" stand for? What is "regular" in such groups? I would like to know idea behind defining these groups, and naming these groups "regular." ...
2
votes
1answer
64 views

Monoid with inversion

Is there a name for monoid with operation $a\mapsto a^{-1}$ conforming the equations $(a^{-1})^{-1}=a$ and $(b\cdot a)^{-1} = a^{-1}\cdot b^{-1}$? (with no requirement that $a^{-1}\cdot a$ would be ...
10
votes
0answers
95 views

Smallest $n$ such that $G$ is a subgroup of the symmetric group $S_n$ [duplicate]

A well-known result, Cayley's theorem, says that any group is isomorphic to $S_n$ for some $n$. Given a (finite) group $G$, is there a standard name for the smallest such $n$? This seems like a very ...
3
votes
2answers
105 views

Who did first use the term “simple” in group sense and why?

Who did first use the term "simple" in group sense? More appreciated if I learn the original word(might be in French, Galois???)... and also... Why do you think that this term is chosen ...
2
votes
1answer
51 views

Name for a certain class of groups that contains all the abelian groups

I cam across this type of groups. Is there a name for groups that satisfy this condition: $$\forall x,y\in G[xyx^{-1}\in \langle y\rangle]$$ As mentioned in the title, it is easy to see that all the ...
0
votes
2answers
48 views

Terminological question on “action factors through”

What does it mean that the action of a group on some space factors through the action of another one?
6
votes
2answers
176 views

What is this method called - Abelization of a Group.

Today, I wanted to make a post for this question. There are some approach in which we can overcome the problem like this and this. According to my knowledge, I could solve the problem via the approach ...
3
votes
1answer
90 views

writing $M : \Gamma_{n,0} \backslash \Gamma_n$

Let $\Gamma_n = \operatorname{Sp}_n(\Bbb Z)$ and $\Gamma_{n,0} \subset \Gamma_n$ be a subgroup. We write $$ M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ ...
2
votes
1answer
123 views

What is the reason for the name *left* coset?

Let $G$ be a group and let $H \leq G$ be a subgroup. It seems that it is now standard to call the cosets $$gH=\{gh \ | h \in H \}$$ the left cosets of $H$ in $G$. I have to admit to being slightly ...
4
votes
2answers
102 views

Classes and Sets

Why are equivalence classes called so and not equivalence sets? I am kind of not able to find the difference between a class and a set. What properties that a set have that a class cannot have? It ...
5
votes
1answer
236 views

Terminology for infinite groups, all of whose subgroup have finite index.

Is there a name for (infinite) groups such that every non-trivial, proper subgroup has finite index (e.g. $\mathbb{Z}$)?
1
vote
2answers
62 views

Confusing with the concept of normalizer $N_G(H)$

I'm Confusing with the concept of normalizer $N_G(H)$. It's a stupid question, sorry I'm new in this subject. Following the Hungerford's concept: If $H$ acts by conjugation on the set $S$ of all ...
2
votes
2answers
72 views

What to call $x^{-1}ax$?

If $G$ is a group and $a,x\in G$, then would we call $x^{-1}ax$ a conjugate of $x$ or a conjugate of $a$? Sorry for such a short question, was just doing a problem and want to call this something so ...
2
votes
1answer
89 views

Properties which are constant on conjugacy classes of a group

Let $\Phi$ be some property which might hold of an element of a group, and say that in every group, $\Phi$ holds for some element $x$ of the group if and only if it holds for all the conjugates ...
2
votes
1answer
89 views

Direct product of groups in categorical terms

In the category of abelian groups $\mathsf{Ab}$ the coproduct is the "direct" product (only finite support), but the product is the "cartesian" product (may be infinite support). In the category of ...
2
votes
1answer
70 views

What to call this binary operation property?

In one of my projects I came across a property that all the binary operations need to have to lead to correct results. It looked like a general property so I was wondering if it might have been ...
6
votes
1answer
73 views

If $S_n$ is a symmetric group, what is the name for $n$?

If $S_n$ is a symmetric group, we call $n!$ to be the size/order of a symmetric group. $S_n$ permutes $n$ symbols, what is the name for this number $n$?
2
votes
0answers
40 views

Usage and determination of “rank” and “dimension” of groups & representations

Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately. A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra ...
2
votes
1answer
129 views

What does “lifted action” mean?

I read about angular moment and linear moment but I don't know what "lifted action" means. Can you explain please? Thanks. :)
3
votes
4answers
218 views

Name that theorem: $G/\ker(\phi) \cong \bar{G}$ for epimorphism $\phi : G \to \bar{G}$

Theorem 2.7.1 in Topics in Algebra 2$\varepsilon$ by I.N. Herstein goes Theorem 2.7.1. Let $\phi$ be a homomorphism of $G$ onto $\bar{G}$ with kernel $K$. Then $G/K \cong \bar{G}$. The preceding ...
8
votes
1answer
112 views

If $H$ is a subgroup of $G$ and $x,y\in G$, what is $xHy$ called?

For a group $G$, its subgroup $H$ and $x,y\in G,$ we call $xH$ a left coset of $H,$ and we call $Hy$ a right coset of $H.$ Is there a special name for sets of the form $xHy$? Is there a name or ...
26
votes
3answers
743 views

Where does the word “torsion” in algebra come from?

Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry ...
1
vote
1answer
98 views

What group does $\mathbb{G}_m$ denote?

What group does $\mathbb{G}_m$ denote? I saw it used here.
5
votes
4answers
1k views

What is the difference between a Subgroup and a subset?

What is the difference between a Subgroup and a subset? I know hardly any Abstract algebra, just some things from youtube and wikipedia, but the notion of a subgroup being part of a larger group and a ...
1
vote
3answers
434 views

Term for a group where every element is its own inverse?

Several groups have the property that every element is its own inverse. For example, the numbers $0$ and $1$ and the XOR operator form a group of this sort, and more generally the set of all ...
1
vote
1answer
47 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 ...
3
votes
1answer
99 views

Motivation for the term “transitive” group action

I have two questions: In a text, I read that a group permutes pairs of faces of a solid transitively. Geometrically, what are they referring to, and what is an example of when a group may not ...
1
vote
1answer
53 views

Definition of tautological action

What is the precise meaning of the term 'tautological action' as used for example in this Wikipedia page in the context of semigroup actions? For reference the particular sentence is: "A ...
2
votes
2answers
114 views

Terminology: $H$ and $K$ are subgroups. What is $HK$ called?

Let $H, K\leq G$. I was wondering what you call the "product" $HK$ of $H$ and $K$. I was trying to verbalise the steps of showing $G$ is a semidirect product: Normality of $H$: $H\unlhd G$. Trivial ...
1
vote
2answers
337 views

Automorphism groups and symmetric groups

Looking on the Wikipedia page for automorphism; in the examples it first states that in set theory, the automorphism of a set $X$ is an arbitrary permutation of the elements of $X$, and these form the ...