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Mathematics Stack Exchange

Questions tagged [2-groups]

For questions regarding groups of even prime power order, as distinct from p-groups in general. Topics include 2-groups of maximal class, 2-groups as Sylow subgroups, and the conjecture that almost all groups are 2-groups. Not intended for use with the p-groups tag.

$2$-groups are undoubtedly special among $p$-groups, set apart by many strange and often troublesome qualities. This tag is intended to collect questions about what makes $2$-groups special, how $2$-groups work differently from other $p$-groups, and the inner workings of specific families of $2$-groups. Other topics include $2$-groups of maximal class, $2$-groups as Sylow subgroups, and the conjecture that almost all groups are $2$-groups.

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References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to full blown analogs proved with vastly…
Alexander Gruber
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17
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2 answers

What do Sylow 2-subgroups of finite simple groups look like?

What do Sylow 2-subgroups of finite simple groups look like? It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision many answers. I tried to give an answer of "broad…
Jack Schmidt
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10
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1 answer

Proportion of nonabelian $2$-groups of a certain order whose exponent is $4$

Let $$\displaystyle A(n)=\frac{\text{number of nonabelian 2-groups of order $n$ whose exponent is }4}{\text{total number of nonabelian 2-groups of order $n$}}.$$ Using GAP, I could observe the following: $$A(16)=\frac{5}{9}=0.5556,…
Chuks
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7
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1 answer

On $2$-groups with a property

If $G$ is a non-abelian $p$-group ($p>2$) such that any two maximal cyclic subgroups have trivial intersection, then $G$ is of exponent $p$ (see "Groups of Prime Power Order-1"- Berkovich, Exer. 2, p.27). Here, "maximal cyclic subgroups" means…
Beginner
  • 11,164
6
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1 answer

Abelian subgroup in a 2 group.

Let $G$ be a non-abelian 2-group of order greater than or equal to 32 and $|Z(G)|=4$. Does the group $G$ has an abelian subgroup $H$, such that $16 \leq |H| \leq |G|/2$?
D. N.
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6
votes
1 answer

Why are there lots of groups with order $2^n$?

I've noticed that, in the OEIS sequence A000001, lots of record high values are held by powers of $2.$ The records are held by only $1, 4, 8, 16, 24, 32, 48, 64, 128, 256, 512,$ and $1024.$ The only record holders that aren't powers of two are $24$…
mathlander
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3
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0 answers

Characterize all non abelian $2$-groups which has no subgroup isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$.

We already know that for any odd prime $p$, every non abelian $p$-group contains a subgroup isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$. So can we say something for even prime? I observed that if $G$ is a non abelian group of order $2^n$ with…
Black Widow
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$2$-groups with odd permutations

If $P$ is the Sylow $2$-subgroup of a finite group $G$, $H
user151882
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2
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1 answer

Can we find a non central element of order 2 in a specific 2-group?

Let $G$ be a non-abelian group of order $2^5$ and center $Z(G)$ is non cyclic. Can we always find an element $x\not\in Z(G)$ of order $2$ if for any pair of elements $a$ and $b$ of $Z(G)$ of order $2$, the factor groups $G/\langle a\rangle$ and…
user93432
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2
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1 answer

Minimal number of relations in finite 2-groups with 2, 3, and 4 generators

I have received helpful answers to my two previous questions that focused on the symmetric group of degree 3 and the dihedral group of order 8. If $d$ is the minimal number of generators of a finite group, and $r$ is the minimal number of…
Elliot Benjamin
  • 187
1
vote
1 answer

on automorphisms groups a finite 2-group

Let $G=\langle a,b\mid a^4=b^4=1, bab^{-1}=a^3\rangle$. Please prove that $Aut(G)$ is generated by the automorphisms $$a\mapsto ab,\hspace{10pt} a\mapsto a^3,\hspace{10pt} a\mapsto ab^2, \hspace{10pt}b\mapsto a^2b…
elham
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Extensions of classifying spaces and higher-groups

$\newcommand{\B}{\mathrm{B}}\newcommand{\U}{\mathrm{U}}$The extension of a group $G$ by $\B\U(1)$, where $\B$ denotes the classifying space specifies a 2-group, $\tilde{G}_2$ $$ 1\to\B \U(1)\to \tilde{G}_2\to G \to 1. $$ Similarly, extending by…
ɪdɪət strəʊlə
  • 283
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Finite $2$-groups of order $>32$ and nilpotency class $2$

I only know $2$-groups of nilpotency class $2$ and order less than or equal to $32$, and wondering if there are finite $2$-groups of order $>32$ and nilpotency class $2$? Your suggestions are appreciated. Thanks.
Chuks
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0
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Quaternion group as a central extension of a $2$-group

I'll appreciate it if you help me to tackle this situation. I'm going to characterize 2-group $G$ whose two main properties such as $cd(G)=\{1,2\}$ and there is normal abelian subgroup $P$ such that $G/P \cong Q_8$.
khatoon khedri
  • 77
0
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1 answer

Direct Product of Chernikov Groups is Chernigov group?

A group $G$ is said to be Chernikov if it contains a normal subgroup N such that $G/N$ is finite and $N$ is direct product of finitely many Prufer groups. The problem is the following: If $G$ is a $2$-group and direct product of finitely many…
Carmine Monetta
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