Japanese journal of mathematics. New series
Online ISSN : 1861-3624
Print ISSN : 0289-2316
The stable behavior of the augmentation quotients of some groups of order p4, I
Kazunori HORIBEKen-Ichi TAHARA
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Volume 10 (1984) Issue 1 Pages 137-157

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Abstract

We list the stable structure of these six groups below.
(1) G1=Zp׋x, y, z|xp=yp=zp=[x, z]=[y, z]=1, [x, y]=z›,
n0=3p-2, π=2
Q3p-2(G1)=Zp(1/2)(p+1)(p2+p+1), Q3p-1(G1)=Zp(1/2)(p+1)(p2+p+1)+1
(2) G2=‹u, x, y|up=xp=yp2=[u, y]=[x, y]=1, [u, x]=yp›,
n0=3p-2, π=1,
Q3p-2(G2)=Zp(2p2+p+1).
(3) G3=Zp׋x, y|xp=yp2=1, [x, y]=yp›,
n0=3p-2, π=1,
Q3p-2(G3)=Zp(2p2+p+1).
(4) G4=Zp(2)×Zp2
n0=3p-2, π=1,
Q3p-2(G4)=Zp2×Zp(2p2+p-1).
(5) G5=‹u, x, y|up=xp=yp2=[u, x]=[u, yp]=1, [u, y]=x, [x, y]=yp›,
n0=4p-3, π=2,
Q4p-3(G5)=Zp(1/2)(3p2+1)+p, Q4p-2(G5)=Zp(3/2)(p2+1)+p
(6) G6=‹u, x, y|up=xp=yp2=[u, x]=[x, y]=1, [u, y]=x›,
n0=4p-3, π=2,
Q4p-3(G6)=Zp2×Zp(3/2)(p2-1)+p, Q4p-2(G6)=Zp2×Zp(1/2)(3p2-1)+p.

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