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.

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votes
1answer
149 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 $2$,...
10
votes
1answer
242 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
vote
0answers
37 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 ...
0
votes
1answer
41 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
1answer
65 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 ...
2
votes
0answers
51 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 ...
1
vote
1answer
77 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{...
12
votes
2answers
838 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 ...
6
votes
1answer
163 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 $16 \leq |H| \leq |G|/2$?
7
votes
1answer
172 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, ...
26
votes
1answer
315 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 ...