20
votes
3answers
622 views

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 ...
0
votes
0answers
68 views

What does $\langle A,B\rangle$ mean?

If $G$ is a group and $A,B$ are subgroups of it (or, I guess the definition just needs $A$ and $B$ to be subsets of $G$), what does $\langle A,B\rangle $ mean? I know what $\langle A\rangle $ means ...
2
votes
1answer
29 views

Group order notation?

I'm working with Dummit and Foote's Abstract Algebra text, and I encountered some notation that confused me. The theorem I saw it in reads as follows. Suppose $\varphi:G \rightarrow H$ is a ...
1
vote
1answer
17 views

Notation in modulo groups

What does ${\Bbb Z}_m^*$ mean? I know that $\Bbb Z_m$ is isomorphic to $\Bbb Z/m \Bbb Z$ but the asterisk tripped me up.
1
vote
1answer
24 views

Notation question: Group generated by two elements?

Let there be $H$ subgroup of symmetric group $S_4$, so that $H= \langle (12)(34),(234) \rangle$. What does the notation $\langle (12)(34),(234) \rangle$ mean? I know that if there's one elements, then ...
1
vote
0answers
26 views

Groups - Compositions

If the f is written to the right of its argument does that mean the composition of $f g$ is actually $g(f(x))$ instead of being $f(g(x))$ which is the notation I'm used to. I ask this because I read ...
0
votes
1answer
32 views

Groups/Sets Notation Question

Simple question: But what does the sigma small Y mean, does it just represent a group? Also have seen this with numbers, and not quite sure what it means. Thanks
1
vote
1answer
44 views

How to compute cyclic notation (23)(1)

I seem to become confused whence computing simple cyclic notations as such. From my understanding, the rule goes by starting from the right and to the left. However by doing this I only end up with ...
7
votes
1answer
122 views

Commutator subgroup - or?

If $G$ is a group and $X, Y \subseteq G$ then the commutator subgroup of $G$ is defined as $[G, G] = \langle [x, y] \mid x, y\in G \rangle$, where $[x, y] = x^{-1}y^{-1}xy$ and the group generated by ...
6
votes
2answers
109 views

Group Theory - Quotient group notation?

What is the difference between the following terms: $\mathbb{Z}_{4}$ , $\mathbb{Z}/4$ and $\mathbb{Z}/{4}\mathbb{Z}$ ? I am pretty sure the first one is the cyclic group with addition modulo 4... ...
1
vote
0answers
39 views

Notation for pointwise versus “setwise” stabilizers

Suppose one is working with both pointwise and setwise stabilizers of sets under a group action. Are there common conventions for notationally distinguishing these two notions? How common are they? ...
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 ...
0
votes
2answers
54 views

Definition of $[G:C_G(x)]$

What is the meaning of $[G:C_G(x)]$ in group theory? Is this equivalent to $\frac{|G|}{|Z_G(x)|}$, or to $|Z_G(x)|$?
3
votes
2answers
142 views

Group theory notation

What does the notation $(G,.)$ mean in group theory? I have seen in places that $.$ implies the binary operation multiplication on group $G$. But then, why do we show an abelian group as $(G, +)$? And ...
2
votes
2answers
101 views

The notation $\mathbb{Z}[\alpha]$

Let $\alpha$ be a real number. I'm studying group theory from the notes of my brother (I'm 16) and I often jump into the notation $\mathbb{Z}[\alpha]$, which, however, is defined nowhere through the ...
2
votes
1answer
114 views

Notation in group theory?

I have three questions on notation. What does the squiggly line mean in the following: $Aut(G) \cong Aut(G) \wr \mathbb{Z}_2$ What does $\rtimes$ mean in the following: $ \varphi :(Aut(G)\times ...
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
1answer
39 views

Using additive or multiplicative notation when showing isomorhphism (between a subgroup and its left coset)?

I'm trying to understand the proof of that all left cosets of a group G with respect to a subgroup H are equivalent. http://www.proofwiki.org/wiki/Cosets_are_Equivalent And I'm stuck on a tiny ...
1
vote
2answers
186 views

Do you know this notation in group theory?

Somebody know this notation in group theory: $$X^G,$$ where $G$ is a group and $X$ aparently is a subset of G? I've come across with this notations in the following problem: Show that $X^G = ...
0
votes
0answers
44 views

Interpreting Notation (stabilizer)

Let $f$ be a separable polynomial, $H, \subset L \subsetneq S_n$ groups, $L/H=\{ H,...,t_e H \}$, $\Theta \in k[x_1,...,x_n]$ s.t. $\mathrm{Stab}_L(\Theta)=H$, $\theta=\widetilde{\Theta}=$evaluation ...
2
votes
1answer
76 views

What's the difference between $ \mathbb{Z}/4\mathbb{Z}$ and $ 4\mathbb{Z} $?

Can someone please explain the difference between $ \mathbb{Z}/4\mathbb{Z} $ and $ 4\mathbb{Z} $? From my understanding (please correct where I'm wrong): the group $4\mathbb{Z}$ has only four ...
3
votes
1answer
100 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 $$ M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ ...
5
votes
3answers
205 views

What is the meaning of the parentheses in $\phi^{-1}\left[\{\phi(g)\}\right]=gH=Hg$?

I am studying homomorphisms is groups and i saw a theorem saying: For $g$ in a group $G$, the cosets $gH$ and $Hg$ are the same, and collapsed onto the single element $\phi(g)$ by $\phi$. That is, ...
3
votes
2answers
126 views

$H ≤G$ means $H$ is a subgroup of $G$?

I was reading this page: http://www.proofwiki.org/wiki/Definition:Subgroup I never heard that $H ≤G$ means $H$ is a subgroup of $G$. Is this standard notation ? And if not, what is/are normal ...
2
votes
2answers
3k views

What does the plus sign contained in a circle ($\oplus$) mean in this case?

The fundamental group of the torus is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. I know that the $\oplus$ symbol is the exclusive or symbol but I don't understand how two of the same sets are ...
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 ...
0
votes
1answer
98 views

Notations for one-sided coset spaces

Let $G$ be a group and let $H$ be its subgroup. What notations are used for the coset spaces $\{gH\,|\,g\in G\}$ and $\{Hg\,|\,g\in G\}?$ What I've written is a bit much to type, and I will be writing ...
1
vote
1answer
102 views

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

What group does $\mathbb{G}_m$ denote? I saw it used here.
6
votes
1answer
144 views

What does this notation mean: $\displaystyle\lim_{\leftarrow} \,\mathbb{Z}/n\mathbb{Z}$?

Just a small notation question from this Wikipedia page: The absolute Galois group of a finite field $K$ is isomorphic to the group $$\hat{\mathbb{Z}}=\lim_{\leftarrow} \mathbb{Z}/n\mathbb{Z}.$$ ...
5
votes
5answers
348 views

Interpretation of Symbol: “$\rtimes$”

What exactly does $\text{Aff}(2) = \mathbb{R}^2 \rtimes SL_{2}(\mathbb{R})$ mean? I know that it is the group of area preserving affine transformations of (oriented) $\mathbb{R}^2$. But how would you ...
0
votes
1answer
42 views

Is there a common notation for a group formed by appending a unit (eg. seconds) to some group?

Assume $(G,+)$ is an (abelian) group. And we have some unit $u$ (not the unit from ring theory, but some measurement unit like meters, seconds, etc.). Then with $G_u = \{k\ u \mid k \in G\}$ and ...
3
votes
2answers
131 views

Original papers on the subject of group actions

Does anyone if there are any original paper(s) that first introduced the notion of group action or permutation representation, and who the author(s) were? Any references I have found so far on e.g. ...
1
vote
1answer
255 views

Notations in Group theory

I will start by apologizing as many will not like this question. I am reading the paper COHOMOLOGY THEORY OF GROUPS WITH A SINGLE DEFINING RELATION and having focused on typology throughout my ...
1
vote
2answers
109 views

What does $s^t$ mean in group theory?

For subset $S$ and $T$ of a group, define $ST = \{st|s \in S, t \in T\}$ and $S^T = \{s^t|s \in S, t \in T\}$. What does $s^t$ mean in this context?
0
votes
1answer
107 views

Don't understand the notation for this group theory question

Could someone explain the notation in this question to me so I can have a go at answering it. Show that $SO_3(\mathbb{F_2}) = \{M \in SL_3(\mathbb{F_2})|M^{-1} = M^t\}$, where $M^t$ is the transpose ...
0
votes
1answer
95 views

What is the Algebraic Structure $\bf{Z}^*$?

The following is question I am being asked: Find a homomorphism from the group $(D_3, \circ)$ of all symmetries of the equilateral triangle to the group $\bf{Z}^*$. But what algebraic ...
8
votes
1answer
674 views

On 'backslash-forward slash' notation

I am curious about a notation that I have seen, but I have only seen it in contexts beyond my current level of ability and so haven't learned its meaning. Also, it's often difficult to search for the ...
3
votes
1answer
122 views

Name or notation for $\mathbb Z/2\mathbb Z\ast\mathbb Z/2\mathbb Z\ast\cdots\ast \mathbb Z/2\mathbb Z$

Is there a standard notation for the n-fold free product of a group with itself? In particular, I'd like to know a nice name or notation for the the $n$-fold free product of $\mathbb Z/2\mathbb Z$ ...
0
votes
1answer
85 views

Name this property: First symmetric group in which a given group appears

For all finite groups $G$, define $S(G)$ to be the smallest $n\in\mathbb{Z}^+$ such that there exists an $H\leq S_n$ isomorphic to $G$ — i.e., $S(G)$ is the index of the first symmetric group in which ...
0
votes
1answer
114 views

notation for symmetry types

I am reading an article and in one of the sections the article mentions the symmetry group. The symmetry group of one of the objects the article talks about is the dihedral group of order 12, using ...
0
votes
2answers
136 views

How do you write the following in proper notation?

I would like help in determining the proper notation to say: The a group $G$ acting on a set of 3 points formed by the quotients $G/H$ where $H$ is a normal subgroup of $G$ is homomorphic to $S_3$ ...
4
votes
4answers
479 views

How to read permutation symbols like $(123)$?

I'd be grateful for some help reading permutation symbols such as $(123)$. Does it mean, when applied to a target sequence such as $(x y z w)$, "replace the element in the first slot of the target ...
1
vote
1answer
88 views

Naming: How to call a direct product of elementary abelian groups?

Is there an accepted name for abelian groups of the form $\prod_{i=1}^n \mathbb{Z}_{p_i}$ for some primes $p_1,\dotsc,p_n$? (i.e: direct products of cyclic groups of prime orders, or in other words - ...
2
votes
1answer
202 views

Help with definition of group realization

I am reading a document it says A group realization is a map from elements of G to transformations of a space M that is a group homomorphism, i.e. it preserves the group multiplication law. Thus if ...