MAGMA 2012



Quantum Class Groups

Foundations


To describe the transfer targets and kernels of a quantum class group Gp2(K),
it is necessary to investigate certain quotients G of coclass cc(G) = 1.
First, we consider abelianizations of diamond type (p,p) for an odd prime p ≥ 3.
The proofs of the following two theorems are due to Heider and Schmithals.
However, here we express the statements in a purely group theoretical form
which is convenient to emphasize our intention.


1.1. Coarse Transfer Kernel Type (cTKT)

Theorem 1. Let p ≥ 3 be an odd prime, n ≥ 2 an integer,
and G ∈ CF(n,n;p) a p-group of order |G| = pn and class cl(G) = n-1,
having an abelian maximal subgroup A.
(Consequently, G is of coclass cc(G) = 1 with abelianization G/G' of diamond type (p,p)
and cyclic lower central factors (Gj:Gj+1) = p for 2 ≤ j ≤ n-1.)
Denote the transfer from G to A by T : G/G' → A. Then
1.1. ker(T) = G/G', if and only if G ≅ G0n(0,0),
1.2. ker(T) = A/G', if and only if G ≅ G0n(0,1),
2. ker(T) = M/G', for some maximal subgroup M ≠ A, if and only if G ≅ G0n(gi,0),
where g ≥ 1 denotes the smallest primitive root modulo p and 0 ≤ i ≤ gcd(n-2,p-1)-1.

Remark 1. In case 1.1, T is called a total transfer,
in case 1.2 a partial transfer with fixed point, and
in case 2 a partial transfer without fixed point.
The coarse Taussky type (A) of T includes cases 1.1 and 1.2, characterised by ker(T) ∩ A/G' > 1,
whereas Taussky type (B) coincides with case 2, where ker(T) ∩ A/G' = 1.

Blackburn's metabelian p-groups Gan(z,w) are defined in our paper
Transfers of metabelian p-groups.


1.2. Transfer Target Type (TTT)

Theorem 2. Let p ≥ 3 be an odd prime, n ≥ 2 an integer,
and G ∈ CF(n,n;p) a p-group of order |G| = pn and class cl(G) = n-1,
having an abelian maximal subgroup A of index (G:A) = p and order |A| = pm, m = n-1.
Denote the transfer from G to A by T : G/G' → A.
Then the structure of A is given by
1.1. A ≅ ((pq+1,…,pq+1)r times,(pq,…,pq)(p-1)-r times), where m = q(p-1)+r, q ≥ 1, 0 ≤ r < p-1, if m ≥ p+1,
1.2. A ≅ (p2,(p,…,p)p-2 times), if m = p and T is of type (A),
1.3. A ≅ (p2,(p,…,p)p-2 times), if m = p, T is of type (B), and G not ≅ G0p+1(1,0),
2.1. A ≅ (p,…,p)p times, if m = p, T is of type (B), and G ≅ G0p+1(1,0),
2.2. A ≅ (p,…,p)m times, if 1 ≤ m ≤ p-1 and T is of type (A),
3. A ≅ (p2,(p,…,p)m-2 times), if 2 ≤ m ≤ p-1 and T is of type (B).

Remark 2. In cases 1.1, 1.2, 1.3, A is a nearly homocyclic abelian p-group of p-rank p-1,
in cases 2.1, 2.2 an elementary abelian p-group of p-rank ≤ p,
and in case 3, A has an exceptional structure of p-rank ≤ p-2.

The only case with elevated p-rank p is 2.1, where G ≅ G0p+1(1,0) ≅ SylpA(p2).

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