# Tagged Questions

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### What do $\Bbb N^*$ and $\Bbb Z(p^n)$ mean in this paper?

There is a theorem: in this paper: http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ78_01%2FS1446788700015548a.pdf&code=2ffd5c5100675caf83c2e95bce65491e But there is no explanation ...
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### Question About Group Theory Notation

I am having trouble understanding what "Universal Cover of $\mathbb{Z} \times \mathbb{Z}$" mean exactly. Thanks
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### Compounding unary operators

I am working with the symmetric group $S_5$. I have 3 unary operators defined: $R$, $T$, and $O$, and I'm writing about their composition. Suppose I want to denote the compound operation of "$T$, ...
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### Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
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### What does $\frac12(D_{2p}\times D_{2p})$ mean in group theory?

Reading a thesis, I have come across the (unexplained) notation $$\frac{1}{2}(D_{2p}\times D_{2p})\cong (p\times p):2,$$ where $D_{2p}$ is a dihedral group. What does this "$\frac12$" notation mean? ...
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### On group-theoretic shorthand notation

I have often seen shorthand notation used in group-theoretic contexts and I believe it is called ATLAS notation. However, even with some searching I have not been able to find a satisfactory summary ...
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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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### 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 ...
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### 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 ...
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### 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.
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### Notation question: Subroup 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 ...
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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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### 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... ...
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### 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? ...
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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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### 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 $|Z_G(x)|$?
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### 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 ...
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### 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 ...
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### 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 ...
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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 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, ...
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### $H ≤G$ 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 ...
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### 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 ...
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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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### 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 ...
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### What group does $\mathbb{G}_m$ denote?

What group does $\mathbb{G}_m$ denote? I saw it used here.
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### 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 $$\hat{\mathbb{Z}}=\lim_{\leftarrow} \mathbb{Z}/n\mathbb{Z}.$$ ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb Z/2\mathbb Z$ ...
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### 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 ...
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### 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 ...
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### 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 $S_3$ ...
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### 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 ...
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 - ...