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

(Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other centralizers NOT isomorphic to these two groups):

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

ii. the $H_8 \times Z_2$

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(Q2) Which finite groups $G$ ONLY contain specific centralizers isomorphic to these two groups (but contain NO other centralizers isomorphic to anything else):

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

ii. the $H_8 \times Z_2$

where $H_8$ is the quaternion group with the order of $|H_8|=8$ (be an Hamiltonian). 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 centralizers. (NO need to be complete.) :o)

see also this.

Community
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annie marie cœur
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  • In Q2, $H_8 \times Z_2$ contains an element of order 4 with centralizer isomorphic to $Z_4 \times Z_2$, which is not on the list. Do you mean that these are the only centralizers of involutions? – Jack Schmidt Apr 18 '14 at 20:49
  • @ Jack, yes, that is what I mean. Indeed $H_8$ has 5 conjugacy classes, with 2 centralizers isomorphic to $H_8$, and another 3 centralizers isomorphic to $Z_4$. If so, for Q2, it impossible to have only $H_8 \times Z_2$ centralizers without also including centralizers of $Z_4 \times Z_2$? – annie marie cœur Apr 18 '14 at 21:26
  • @ Jack, If that is the case, then we shall be allowed to include the centralizers of $H_8 \times Z_2$, which is $H_8 \times Z_2$ and $Z_4 \times Z_2$. – annie marie cœur Apr 18 '14 at 21:33
  • What is the motivation for these questions? You say it is a simple question - so why do you need help with it? – Derek Holt Apr 18 '14 at 21:42
  • Maybe it is not simple, I remove it. I am just simply obsessed with the structure of finite groups. (clarify: it is not a HW problem.) – annie marie cœur Apr 18 '14 at 21:49
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    SmallGroup(128,934) satisfies (Q1). But I still don't understand why you are interested in this specific pair of centralizers. – Derek Holt Apr 18 '14 at 22:10
  • this is interesting. What is the order of this group? Can it be further decomposed by a direct product or semi-direct product of elementary groups, cyclic groups or Dihedral, Quaternion groups etc? – annie marie cœur Apr 18 '14 at 22:22
  • p.s. the motivation is programming a rubik cube problem in a higher dimensional space. This finite group structure seems helping writing the program code. (not 100% sure though.) – annie marie cœur Apr 18 '14 at 22:38
  • If you are writing programs involving groups then surely you must use GAP, in which case you can investigate SmallGroup(128,934) for yourself - it has order $128$ of course! – Derek Holt Apr 19 '14 at 07:46
  • let us continue this discussion in chat – annie marie cœur Apr 19 '14 at 19:14
  • Yes GAP, I was trying 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 19:16

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The comments suggest that you mean only centralizers of involutions. Even in that case, no finite group $G$ can have only the two involutionn centralizers you suggest, so Q2 seems to have a negative (or empty) answer. One involution centralizer must contain a Sylow $2$-subgroup. Hence the Sylow $2$-subgroup of $G$ must have order $16$. But the elementary group of order $16$ and the $H_{8} \times Z_{2}$ are then both Sylow $2$-subgroups of $G$, a contradiction, as they are clearly not conjugate.

Geoff Robinson
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  • Thanks Geoff, the negative answer is not surprising. How about this one: http://math.stackexchange.com/questions/759647/which-finite-groups-contain-the-two-specific-centralizers?lq=1; does SmallGroup(128,934) contain $H_8 \times Z_2$ other than $D_4 \times Z_2$ and $(Z_2)^4$ as centralizers? – annie marie cœur Apr 19 '14 at 00:39