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512.542.3

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íÏÖÎÏ ÓËÁÞÁÔØ tth.zip

ÞÉÔÁÊÔÅ readme.txt

Contents

1  ã×ÅÔÁ
2  ëÁÒÔÉÎËÁ
3  Gif
4  ÔÁÂÌÉÃÁ
5  ÔÁÂÕÌÑÃÉÑ
6  éÎÄÅÈ
7  ìÅÍÍÁ
8  íÁÔÒÉÃÙ

1  ã×ÅÔÁ

Red

Magenta

Blue

Cyan

Green

2  ëÁÒÔÉÎËÁ


3  Gif


4  ÔÁÂÌÉÃÁ

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5  ÔÁÂÕÌÑÃÉÑ

qqqqqqqq¯ wwwwwww¯ eeeeeeee¯ q w r
q w r
q w r
q w r

ëÁË ×ÉÄÉÔÅ ÎÅ ÒÁÂÏÔÁÅÔ

6  éÎÄÅÈ

éÎÄÅÈ

7  ìÅÍÍÁ

ìÅÍÍÁ 1 åÓÌÉ G - ÇÒÕÐÐÁ Ó ÔÏÖÄÅÓÔ×ÏÍ (x[y,z])p = xp , x, y Î G, c Î G ¢ É s - ÃÅÌÏÅ, ÔÏ

  1. (xc)p = xp;
  2. exp (G ¢ ) = p, Gp £ Z(G); [2]
  3. (xs,ys) = (x,y)s.

äÏËÁÚÁÔÅÌØÓÔ×Ï.

(1)

(xc)p = xp.

(2) ÷ÚÑ× x = 1, ÉÚ (1) ÐÏÌÕÞÉÍ exp (G ¢ ) = p. éÚ ÒÁ×ÅÎÓÔ×Á (xp)y = (xy)p = (x[x,y])p = xp ÓÌÅÄÕÅÔ Gp £ Z(G) .

(3) ó ÏÄÎÏÊ ÓÔÏÒÏÎÙ (xsys)p = xpsyps(xs,ys), ÎÏ Ó ÄÒÕÇÏÊ ÓÔÏÒÏÎÙ ÄÌÑ ÎÅËÏÔÏÒÏÇÏ c Î G ¢ (xsys)p = ((xy)sc)p = (xy)ps = (xpyp(x,y))s = xpsyps(x,y)s. é ÐÏÓÌÅ ÐÒÉÒÁ×ÎÉ×ÁÎÉÑ É ÓÏËÒÁÝÅÎÉÊ ÐÏÌÕÞÉÍ ÔÒÅÂÕÅÍÏÅ.

8  íÁÔÒÉÃÙ

ìÅÍÍÁ 2 åÓÌÉ a1, ¼ ,ap-1 É b1, ¼ ,bp-1 - ÜÌÅÍÅÎÔÙ ÁÄÄÉÔÉ×ÎÏÊ ÜÌÅÍÅÎÔÁÒÎÏÊ ÁÂÅÌÅ×ÏÊ p-ÇÒÕÐÐÙ É
aj = p-1
å
 i = 1  
 bi ·ji ,        (j = 1, ¼ ,p-1),
ÔÏ
bi = - p-1
å
 j = 1  
 aj ·jp-1-i ,        (i = 1, ¼ ,p-1).

äÏËÁÚÁÔÅÌØÓÔ×Ï. õÓÌÏ×ÉÅ ÌÅÍÍÙ ÍÏÖÎÏ ÚÁÐÉÓÁÔØ × ×ÉÄÅ A = B·M, ÇÄÅ A = (a1, ¼ ,ap-1) É B = (b1, ¼ ,bp-1) - "×ÅËÔÏÒÙ", Á

M = æ
ç
ç
ç
ç
ç
è
11
21
¼
(p-1)1
12
22
¼
(p-1)2
:
:
···
:
1p-1
2p-1
¼
(p-1)p-1
 ö
÷
÷
÷
÷
÷
ø
- ÍÁÔÒÉÃÁ Ó ÜÌÅÍÅÎÔÁÍÉ ÉÚ Z/pZ. ðÕÓÔØ
N = - æ
ç
ç
ç
ç
ç
è
1p-2
1p-3
¼
10
2p-2
2p-3
¼
20
:
:
···
:
(p-1)p-2
(p-1)p-3
¼
(p-1)0
 ö
÷
÷
÷
÷
÷
ø
÷ÏÓÐÏÌØÚÏ×Á×ÛÉÓØ ÔÅÍ, ÞÔÏ
p-1
å
 i = 1  
 ip-1 º -1 mod p, Á p-1
å
 i = 1  
 ir º 0 mod p
ÐÒÉ r\not º 0 mod p-1 [1], ÌÅÇËÏ ÐÒÏ×ÅÒÉÔØ, ÞÔÏ M·N - ÅÄÉÎÉÞÎÁÑ ÍÁÔÒÉÃÁ É, ÚÎÁÞÉÔ, B = A·N. þÔÏ É ÔÒÅÂÏ×ÁÌÏÓØ ÄÏËÁÚÁÔØ.

References

[1]
B.Huppert., Endliche Gruppen I. Springer-Verlag, Berlin, Heidelberg and New York, 1976.

[2]
Avinoam Mann., Regular p-groups II. Israel J.Math. 14(1973) 294-303.

[3]
Avinoam Mann., Regular p-groups . Israel J.Math. 10(1971) 471-477.

[4]
J.R.J.Groves., Regular p-groups and words giving rise to commutative group operations. Israel J.Math. 24(1976), 73-77.
[5]
J.R.J.Groves., On minimal irregular p-groups. J. Austral. Math. Soc. 16(1973) No 1 78-89.

[6]
P.Hall., A contribution to theory of groups of prime power order. Proc. London Math. Soc. 36(1933), 25-95.

[7]
B.Huppert, N.Blackburn., Finite Groups II. Springer-Verlag, Berlin, Heidelberg and New York, 1982.

[8]
÷.íÁÇÎÕÓ, á.ëÁÒÒÁÓ, ä.óÏÌÉÔÅÒ. ëÏÍÂÉÎÁÔÏÒÎÁÑ ÔÅÏÒÉÑ ÇÒÕÐÐ. îÁÕËÁ. íÏÓË×Á 1974.

[9]
Paul M. Weichsel., Just irregular p-groups. Israel J.Math. 10(1971) 359-363.

Index (showing section)

éÎÄÅÈ
     ðÒÉÍÅÒ, 6-0

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