Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,517 questions
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How many copies of the Standard Model gauge group are there in Spin(10)?
Define the Standard Model gauge group to be $\text{S}(\text{U}(2) \times \text{U}(3))$, the subgroup of $\text{SU}(5)$ consisting of block diagonal matrices with a $2 \times 2$ block and then a $3 \...
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A functional equation on $\mathbb{Z}/p\mathbb{Z}$
Let $p$ be a prime such that $2$ is a primitive root of $p$.
We want to find a bijective function $f: (\mathbb{Z}/p\mathbb{Z})^× \to (\mathbb{Z}/p\mathbb{Z})^× $ s.t.
$$f(2k) = f(k) + f(f(k)) $$
$$f(-...
- 121
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Finite groups for which the maximum degree of the prime graph is 2
Does there exist a finite non-solvable and non-almost-simple group satisfying the following conditions?
The degree of every vertex in its prime graph is at most $2$,
If a vertex $p$ in its prime ...
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Linear-algebraic proof of $p$-group fixed-point theorem
Let $P$ be a $p$-group that acts on a finite set $X$. Let $X^P$ denote the $P$-fixed points of $X$. Then:
$$|X|\equiv |X^P|\pmod p$$
There is an easy combinatorial proof of this: $X$ is a disjoint ...
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How close are elements of $\operatorname{SL}_2(\mathbb{Z})$ to symmetric matrices?
This is motivated by my earlier question: Minimal number of variables needed to parametrize $\operatorname{SL}_2(\mathbb{Z})$
A key step in the paper by Leonid Vaserstein cited in the linked question ...
- 30.2k
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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}}
$I have asked this question on MSE, but got no answers or comments, so I decided to try my luck on MO.
Consider a non-connected reductive ...
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Polynomial conditions on cycle space of simple graph
Let $G=(V,E)$ be a simple cubic graph and $\Phi:=\lbrace \phi\rbrace$ a set of cycles such that for every edge $e\in E$ there is a $\phi\in \Phi$ such that $e\in \phi$. We say that $\Phi$ covers $G$. ...
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On a conjecture of A. Shalev
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ be a prime number, $G$ a pro-$p$ group, and $m$ a positive integer. We say that $H(G,m)$ holds if there is a function of $m$ that is an ...
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Search for: natural classes of finite groupoids (and their Cayley graphs) - similar to 15-game
Take a finite group (e.g. $S_n$), take some its elements - generate subgroup - that simple way gives enormous amount of different groups (and actually with generators - hence Cayley graphs).
Question: ...
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Number of subgroups in a semidirect product
Let $G$ be a finite group and $Sub(G)$ denote the number of subgroups of $G$ including the trivial subgroup and $G$ itself. I believe the following is true:
$Sub(H \times K)\leq Sub(H \rtimes K)$ for ...
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Isomorphism type(s) of minimal groups surjecting on both $G$ and $H$
Fix a prime $p$, and two finite $p$-groups $G$ and $H$. Then we can try to find a group $K$ of minimal order such that there exists a surjective homomorphism $\alpha\colon K\to G$ and another ...
- 58.1k
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A character isometry in $A_5$
Background:
Let $G$ be a finite group. Fix a prime $p$. We say that $H\subseteq G$ is strongly $p$-embedded if:
$p$ divides $|H|$;
For every $x\in G-H$, $p$ does not divide $|H\cap H^x|$.
Some facts ...
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A problem on finite groups and automorphisms [closed]
Let $G$ be a finite group and $\phi : G \rightarrow G$ be a group automorphism such that for more than $\frac{3}{4}$ of elements $g \in G$ we have
$$\phi(g) = g^{-1}$$
Prove that for all $g \in G$
$$\...
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Diameter of p-singular Cayley graph of a finite group
Let $G$ be a finite group and $p$ be any prime number dividing $|G|$. Let $S_p(G)$ be the set of all $p$-singular elements of $G$ i.e. elements whose orders are divisible by $p$.
Conjecture: The ...
- 3,955
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Cyclic 2-subgroups of GL(n,Z_p) for n>1
Let $p$ be a prime, $p\neq 2$. Let $Q$ be a cyclic $2$-group and $P$ an elementary abelian $p$-group of rank $n$. Suppose that $Q$ acts faithfully on $P$ (so $Q$ is isomorphic to a subgroup of $GL(n,...
