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

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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=(Zp)4(Passi[6, Theorem 4.7]) n0=3p-2, π=1, Qn0(G)=(Zp)p3+p2+p+1.2) G=(Zp)2×Zp2 (Horibe and Tahara [3, Proposition 4.5]) n0=3p-2, π=1, Qn0(G)=(Zp)2p2+p-1⊕Zp2.3) G=(Zp2)2 (Proposition 4.5) n0=p2+p-1, π=1, Qn0(G)=(Zp)p2-1⊕(Zp2)p+1.4) G=Zp×Zp3 (Tahara and Yamada [11, Proposition 5.6]) n0=3p-2, π=1, Qn0(G)=(Zp)3p-2⊕Zp3.5) G=Zp4 (Passi [6, Theorem 3.1]) n0=1, π=1, Qn0(G)=Zp4.6) G=Zp×‹x, y, z|xp=yp=zp=[z, x]=[z, y]=1, [y, x]=z› (Horibe and Tahara [3, Proposition 3.1]) n0=3p-2, π=2, Qn0+i(G)=(Zp)(1/2)(p+1)(p2+p+1)+i, i=0, 1.7) G=‹x, y, z, w||xp=yp=zp=wp=[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, Qn0+i(G)=(Zp)sn0+i(G) Case 1. p≡1 mod 3, sn0+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, sn0+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|xp=yp=zp2=[y, x]=[zp, x]=1, [z, x]=y, [z, y]=zp› (Horibe and Tahara [3, Proposition 5.3]) n0=4p-3, π=2, Qn0+i(G)=(Zp)(1/2)(3p2+2p+1)+i, i=0, 1.
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