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.
15 questions
3
votes
0
answers
145
views
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 ...
- 161
7
votes
1
answer
961
views
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 ...
- 4,253
1
vote
0
answers
58
views
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, $\tilde{G}_2$
$$ 1\to\B \U(1)\to \tilde{...
- 283
0
votes
1
answer
135
views
Quaternion group as a central extension of a $2$-group [closed]
I'll appreciate it if you help me to tackle this situation.
I'm going to characterize 2-group $G$ whose two main properties such as $cd(G)=\{1,2\}$ and there is normal abelian subgroup $P$ such that $...
2
votes
1
answer
350
views
Can we find a non central element of order 2 in a specific 2-group?
Let $G$ be a non-abelian group of order $2^5$ and center $Z(G)$ is non cyclic. Can we always find an element $x\not\in Z(G)$ of order $2$ if for any pair of elements $a$ and $b$ of $Z(G)$ of order $2$,...
- 587
10
votes
1
answer
291
views
Proportion of nonabelian $2$-groups of a certain order whose exponent is $4$
Let
$$\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 $n$}}.$$
Using GAP, I could observe the ...
- 1,261
1
vote
0
answers
67
views
Finite $2$-groups of order $>32$ and nilpotency class $2$
I only know $2$-groups of nilpotency class $2$ and order less than or equal to $32$, and wondering if there are finite $2$-groups of order $>32$ and nilpotency class $2$? Your suggestions are ...
- 1,261
0
votes
1
answer
80
views
Direct Product of Chernikov Groups is Chernigov group?
A group $G$ is said to be Chernikov if it contains a normal subgroup N such that $G/N$ is finite and $N$ is direct product of finitely many Prufer groups.
The problem is the following:
If $G$ is a $...
2
votes
1
answer
99
views
Minimal number of relations in finite 2-groups with 2, 3, and 4 generators
I have received helpful answers to my two previous questions that focused on
the symmetric group of degree 3 and the dihedral group of order 8. If $d$ is the minimal number of generators of a finite ...
- 187
2
votes
0
answers
67
views
$2$-groups with odd permutations
If $P$ is the Sylow $2$-subgroup of a finite group $G$, $H <P$, and $x \in P$ so that no non-trivial element of $\langle x \rangle$ conjugates into $H$ (in $G$), and $|P|=|H||x|$, how can I show ...
- 452
1
vote
1
answer
105
views
on automorphisms groups a finite 2-group
Let $G=\langle a,b\mid a^4=b^4=1, bab^{-1}=a^3\rangle$.
Please prove that $Aut(G)$ is generated by the automorphisms
$$a\mapsto ab,\hspace{10pt} a\mapsto a^3,\hspace{10pt}
a\mapsto ab^2, \hspace{...
- 1,195
17
votes
2
answers
3k
views
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 ...
- 57.3k
6
votes
1
answer
202
views
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 $16 \leq |H| \leq |G|/2$?
- 2,221
7
votes
1
answer
227
views
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, ...
- 11.2k
26
votes
1
answer
398
views
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 ...
- 28.3k