1. SMALL 4-QUASINORMAL SUBGROUPS

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A subgroup H of a group G is QUASINORMAL if HK = KH for all
subgroups K of G, i.e. ⟨H, K ⟩ = HK. Write H qn G.
In finite groups, quasinormal subgroups are always subnormal.
If G is finite and H qn G, then H /HG lies in the hypercentre of G/HG .
Quasinormal subgroups of finite p – groups can be a rarity.
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Say a subgroup H of a group G is 4-quasinormal (H qn4 G) if ⟨H, K ⟩ = HKHK for all cyclic subgroups K of G.
Every subgroup of a nilpotent group of class 2 is 4-qn.
Consider H = ⟨h ⟩ qn4 G , a finite p –group with p odd.
When H is cyclic and quasinormal in G (p ≥ 3), then
(i) [H, G ] is abelian and
(ii) H acts on [H, G ] as a group of power automorphisms.
If H is quasinormal & of order p , then Hᴳ is elementary abelian.
What happens when H is 4-quasinormal & of order p ?
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From now on, suppose that H = ⟨h ⟩ qn4 G and that |H | = p (≥3). If G has exponent p , then for all g ϵ G ,
⟨h , g ⟩ is either ELEMENTARY ABELIAN of rank ≤ 2 or EXTRASPECIAL of order p 3(Ep ).
So h commutes with [h , g ] and therefore with h g. Thus
H G is ELEMENTARY ABELIAN (*)
Let p ≥ 5. For arbitrary G it can be shown that for all g ϵ G ,
g p commutes with h. Thus G p centralizes H and hence also H G.
By (*), H G is elementary abelian modulo G p. Also
H G ∩ G p lies in the centre of H G and therefore H G has class ≤ 2.
Then H G is regular and hence has exponent p.
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Let ⟨h ⟩, ⟨c ⟩ and ⟨g ⟩ be cyclic groups of order p , p and p ²
respectively. Then let
G = ⟨h ⟩ ⋉ {⟨c ⟩ × ⟨g ⟩},
where [h , g ] = c and [c , h ] = g p. Then H qn4 G and H G ≅ Ep .
But is [H , G ] always abelian?
Let G = ⟨ h , k1, k2 ⟩ be a finite p – group (p ≥ 5) with H = ⟨h ⟩ of
order p and 4-qn in G. Is [[h , k 1] , [h , k2]] = 1?
Let K = ⟨k1, k2 ⟩. When K ≅ C p × Cp , then |G | ≤ p 5 & G has class ≤ 2.
Therefore YES!
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Now let K = ⟨k1, k2 ⟩ ≅ Ep . Then |G | ≤ p 7 & G has class ≤ 3.
Therefore G is metabelian. Therefore again YES!
If K is abelian, then K p lies in the centre of G ; and G /K p has
class ≤ 2. So again G has class ≤ 3 and YES!
The answer is also YES when
(i) |k 1| = p ; or
(ii) K/K p is abelian ; or
(iii) K /K p is extraspecial.
Let K /K p have class 3. Then ?
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