# Tagged Questions

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### Is it “group axiom” or “group definition”?

Some text books of group theory use "group definitions" when introducing group, and some other text books use "group axioms". But it is obvious that terms "definition" and "axiom" are different. Which ...
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### Who named “Quotient groups”?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation ...
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Sometimes I see in books the term "additive abelian groups". In my opinion, when we use addition to represent the group operation, we already have in mind that the operation is commutative. So ...
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### Some basic terms from finite group theory, normalising and centralising

In a proof a read the following "Since $H$ and $N$ normalize each other, if follows that $H \subseteq C_G(N)$". I thought normalising just means that one subgroup is normal with respect to the other, ...
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### Why are groups “abelian” but rings “commutative”?

I have never seen, in any text, a ring whose multiplication is commutative being called an "abelian ring", even though this would make perfect sense, because this term would necessarily refer to ...
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### Can we define the normal set without $G$ being a group?

Let $X$ be a set in $G$ and $G$ be a group. A normal set is a set $X$ for which $gxg⁻¹∈X$ for every $x∈X,g∈G$. It's just like the normality condition for subgroups, except that $X$ doesn't have to be ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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': G/K = G' ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### $G$-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 $X$ ...
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### 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." ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 $$M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ ...
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### 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 ...
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### 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 ...
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### 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. $\mathbb{Z}$)?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n$?
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### 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 ...
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### 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. :)
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### 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 ...
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### 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 ...
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### 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 ...
### What group does $\mathbb{G}_m$ denote?
What group does $\mathbb{G}_m$ denote? I saw it used here.