Questions tagged [group-theory]
For questions about the study of algebraic structures consisting of a set of elements together with a well-defined binary operation that satisfies three conditions: associativity, identity and invertibility.
40 questions from the last 30 days
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Tricks for Computing the Center of a Group
I was doing a homework question about computing the center of a group, and realized everytime I've ever computed the center, I am very explicitly writing down elements and finding restrictions.
I ...
- 349
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Quotient of $\mathrm{GL}_2(\mathbb{C})$ by a finite group
Let us consider the algebraic group $G=\mathrm{GL}_2(\mathbb{C})$ and consider the $S_2$-action given by conjugation with $P_0=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, that is, the $S_2$-...
- 181
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A surjective homomorphism of $\mathbb R$-groups that is not surjective on $\mathbb R$-points.
$\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
$Consider a non-connected reductive group $G$ over the field $\R$ of real numbers.
Write $S=G^0$ for the identity component of $G$, and ...
- 1,408
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2
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Formal and intuitive definitions of nilpotent group
I'm trying to learn the concept of nilpotent groups. On the one hand, there's this formal definition:
$Z_0(G)=1, \; Z_{i+1}(G)/Z_i(G) = Z(G/Z_i(G))$. Least $i$ for which $Z_i(G) = G$ (if exists) is ...
- 448
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Schur's lemma for $S_3$ discrete group [closed]
If we consider the discrete group $S_3$, for which we write the 3DF (unitary) matrix representation. One can reduce this representation to a sum of irreducible representations. This means, that one ...
- 367
2
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1
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74
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Quick questions about left/right group actions [closed]
Background:
Definition: We say the group $G$ acts on a set $X$ if there is a homomorphism $\sigma:G\to S_X.$ Thus $\sigma(G)$ is a subgroup of $S_X,$ the group of all permutations of $X.$ ...
- 4,087
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1
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Cyclicity of the multiplicative group of the integers modulo a prime
Edit. It seems that the Lemma 2 needs already the existence of a primitive root modulo $p$. If there's no other way to prove it, then my argument is pointless.
(NB: I'm aware that there's plenty of ...
- 5,601
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1
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Constructing a group $(G, \Delta)$ isomorphic to $(G, \ast)$. [duplicate]
I encountered this interesting result from the other day and I am struggling to prove it myself.
Proposition: Let $(G, \ast)$ be a group and $\sigma\in \text{Sym}(G)$. Define a binary operation $\...
- 1,221
3
votes
1
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Invariant trees of outer automorphism of free groups
Gaboriau-Jaeger-Levitt-Lustig (Theorem II.1) constructed an invariant $\mathbb{R}$-tree $T$ with $F_n$ action given any outer automorphism of free groups $\Phi\in \mathrm{Out}(F_n)$. They showed that $...
- 923
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1
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Automorphism group of $UT(3,p)$
Here $p>2$ is a prime, the group $UT(3,p)$ is the group of $3\times3$ upper unitriangular matrices with coefficients in $\mathbb{F}_p$:
$$\begin{bmatrix}
1&x&y\\
0&1&z\\
0&0&...
- 151
5
votes
1
answer
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Center of a finite perfect group
This question is distantly related to the following MathStack post: How "big" can the center of a finite perfect group be?
The above post and its answers comment on the size of the center of ...
- 917
3
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Name of the algebraic structure
I am looking for the name of an algebraic structure that generalizes the concept of a monoid. Suppose we have a set $ S $ with a binary operation $ + $ and binary operation $ * $ . They are both ...
- 97
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2
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Is the isomorphism $(G/N_1)/(N_2/N_1)\cong G/N_2$ in the third isomorphism theorem actually an equality?
I tried to understand the concept that, for a group $G$ and two normal subgroups $N_1,N_2$ with $N_1\subseteq N_2$ it holds that
$$(G/N_1)/(N_2/N_1)\cong G/N_2,$$
but my following reasoning seems to ...
-1
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0
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How applicable are the isomorphism theorems? [closed]
How broadly do the (four?) isomorphism theorems apply? Do they hold only of groups? What do they look like in set theory?
- 135
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Quotient group technicality
I'm studying linear Algebra by myself right now and I'm currently reading about quotient groups. I wanted to disprove the following: for two normal subgroups $N_1\subseteq N_2$ of $(G,\cdot)$, $G/N_2$ ...