The stable behavior of the augmentation quotients of some groups of order p⁴, III

抄録

In this section we list up the stable behavior of the augmentation quotients for all p-groups of order p⁴, p an odd prime.
1) G=(Z_p)⁴(Passi[6, Theorem 4.7])
n₀=3p-2, π=1, Q_n0(G)=(Z_p)^p3+p2+p+1.
2) G=(Z_p)²×Z_p² (Horibe and Tahara [3, Proposition 4.5])
n₀=3p-2, π=1, Q_n0(G)=(Z_p)^2p2+p-1⊕Z_p².
3) G=(Z_p²)² (Proposition 4.5)
n₀=p²+p-1, π=1, Q_n0(G)=(Z_p)^p2-1⊕(Z_p²)^p+1.
4) G=Z_p×Z_p³ (Tahara and Yamada [11, Proposition 5.6])
n₀=3p-2, π=1, Q_n0(G)=(Z_p)^3p-2⊕Z_p³.
5) G=Z_p⁴ (Passi [6, Theorem 3.1])
n₀=1, π=1, Q_n0(G)=Z_p⁴.
6) G=Z_p׋x, y, z|x^p=y^p=z^p=[z, x]=[z, y]=1, [y, x]=z›
(Horibe and Tahara [3, Proposition 3.1])
n₀=3p-2, π=2, Q_n0+i(G)=(Z_p)^{(1/2)(p+1)(p2+p+1)+i}, i=0, 1.
7) G=‹x, y, z, w||x^p=y^p=z^p=w^p=[w, x]=[w, y]=[w, z]=1 [z, y]=1, [y, x]=z, [z, x]=w›
(G. Losey and N. Losey [4, Proposition 3.2])
n₀=4p-3, π=6, Q_n0+i(G)=(Z_p)^s_n0+i(G)
Case 1. p≡1 mod 3,
s_n0+i(G)=K(i=0, 4), K+1(i=1, 2, 3), K+2(i=5),
where K=1+p+_??_(p²-1)+_??_(p³-1),
Case 2. p≡2 mod 3,
s_n0+i(G)=L(i=0), L+2(i=1, 5), L+1(i=2, 3, 4),
where L=1+p+_??_(p²-1)+_??_(p³-2).
8) G=‹x, y, z|x^p=y^p=z^p2=[y, x]=[z^p, x]=1, [z, x]=y, [z, y]=z^p›
(Horibe and Tahara [3, Proposition 5.3])
n₀=4p-3, π=2, Q_n0+i(G)=(Z_p)^{(1/2)(3p2+2p+1)+i}, i=0, 1.