2

This is a question requiring the good knowledge of group theory:

(Q1) Which finite groups $G$ contains some specific centralizers both of these two groups:

i. the elementary group $Z_2^4$, and

ii. the $D_4 \times Z_2$

-

(Q2) Which finite groups $G$ ONLY contains specific centralizers isomorphic to these two groups (but contain NOTHING else):

i. the elementary group $Z_2^4$, and

ii. the $D_4 \times Z_2$

where $D_4$ is dihedral group with the order of $|D_4|=8$. Here $Z_2$ is the cyclic group of the order $|Z_2|=2$.

Let us consider the $|G|$ be as small as possible. Your answer only needs to provide just AN example, the order of the group |G| and its all list of centralizers. (NO need to be complete.) :o)

see also this.

Community
  • 1
annie marie cœur
  • 3,857
  • see also this question: http://math.stackexchange.com/questions/759668/which-non-abelian-finite-groups-contain-the-two-specific-centralizers-part-ii – annie marie cœur Apr 18 '14 at 21:27
  • For Q2: $D_4$ contains centralizers $D_4$, $Z_2\times Z_2$ and $Z_4$. If so, we are allowed to include the centralizers of $D_4 \times Z_2$ as $D_4 \times Z_2$, $Z_2 \times Z_2 \times Z_2$ and $Z_4 \times Z_2$. – annie marie cœur Apr 18 '14 at 21:35
  • 1
    SmallGroup(64,138) satisfies (Q1). – Derek Holt Apr 18 '14 at 21:54
  • Thanks Derek, this is interesting. I have troubled to access GAP so I can only learn that: SmallGroup (64,138) = (((C4 x C2) : C2) : C2) : C2; but what is the form of this group? Can it be further decomposed by a direct product or semi-direct product of elementary groups, cyclic groups or Dihedral groups etc? – annie marie cœur Apr 18 '14 at 22:14
  • And what is the order of SmallGroup (64,138)? (the number of group elements?) – annie marie cœur Apr 18 '14 at 22:24
  • @ Derek, also what are all the centralizers of SmallGroup (64,138)? (there are 16 of them, correct?) – annie marie cœur Apr 18 '14 at 22:30
  • You should learn to do those calculations yourself in GAP. – Derek Holt Apr 19 '14 at 16:11
  • I was trying GAP but I have trouble installing it. Sorry that if you can still fill in this: http://math.stackexchange.com/questions/760827/conjugacy-classes-and-centralizers-of-a-smallgroup – annie marie cœur Apr 19 '14 at 19:15
  • Please contact GAP Support to get some help with installing GAP. Describe your operating system and where the installation fails. For such computational questions the only way to proceed is to install it and learn yourself. As for the SmallGroup(m,n), the 1st argument is the order of a group. – Olexandr Konovalov Apr 20 '14 at 19:34

0 Answers0