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\pretitle{ \centerline{\bf { ����� ���� ����. ��⥬�⨪�. ��.~2}}
\centerline{\bf {�������, 1999}}

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\centerline {\bf �.~�.~����஢᪨�, �.~�.~��䮭���, �.~�.~���㡥��}
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%\leftheadtext{\it �.~�.~����஢᪨�, �.~�.~��䮭���, �.~�.~���㡥�� }
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%�.�.����஢᪨�, �.�.��䮭���, �.�.���㡥��
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\title
\centerline{\bf ��㯯� �����஬�� ���ᮡ��} 
\centerline{\bf ���䨣��権 $\leq 7$ ����� ����࠭�⢠ $\Bbb RP^3$}
\endtitle

\abstract \nofrills
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����뢠���� ��㯯� �����஬�� ���ᮡ�� ���䨣��権 $\leq 7$ �����
����࠭�⢠ $\Bbb RP^3$.

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\head
{\bf
1. �������� }
\endhead

{\it ����冷祭���\/} (ᮮ⢥��⢥��� {\it ��㯮�冷祭���\/})
{\it ���ᮡ�� ���\-�\-��\-�\-�\-�� $m$ ����� ����࠭�⢠
$\Bbb RP^3$\/} ���뢠���� 㯮�冷祭�� (ᮮ⢥��⢥���
��㯮�冷祭��) ����� $m$ ����୮ �����ᥪ������ �����
����࠭�⢠ $\Bbb RP^3$. ���⮯�� ⠪��� �����, � �����
���ன ���� ������� ����୮ �����ᥪ��騬��� ���묨,
���뢠���� {\it ���⪮� ���⮯���\/}. ����� ���������⭮��
���䨣��樨 �� �⭮襭�� ���⪮� ���⮯���� ���뢠���� ��
{\it ���⪮-���⮯��᪨� ⨯��\/}.

����� $A$ --- ��㯮�冷祭��� ���ᮡ�� ���䨣���� $m$ �����
����࠭�⢠ $\Bbb RP^3$. ������稬 �१ $\op{Aut}\(A)$ ��㯯�
���⪨� ��⮨��⮯�� ���䨣��樨 $A$. ��㯯� $\op{Aut}\(A)$
������� �� ������⢥ ������⮢ ���䨣��樨 $A$. �� ����⢨�
������ �祢���� �������䨧� ��㯯� $\op{Aut}\(A)$ � ᨬ��������
��㯯� $\bold S_m$. ��ࠧ �⮣� �������䨧�� ���뢠���� {\it ��㯯��
�����஬��\/} ���䨣��樨 $A$.

�.�.~��� \cite{1}, �.�.~����஢᪨� \cite{3,6}, �.~��஡�� �
����஢᪨� \cite{5} ����稫� ���⪮-���⮯����� �����䨪���
��㯮�冷祭��� ���ᮡ�� ���䨣��権 $m$ ����� ����࠭�⢠
$\Bbb RP^3$ ��� $m \leq 5$, $m=6$ � $m=7$ ᮮ⢥��⢥���. �
\cite{4} ����஢᪨� � �.�.~������� �뫨 ������� ����室��� �
������� �᫮��� ���⪮� ���⮯���� 㯮�冷祭��� ���ᮡ��
���䨣��権 $m$ ����� ����࠭�⢠ $\Bbb RP^3$ �� $m \leq 7$.
��, � ��⭮��, �������� ������ ��㯯� �����஬��
��㯮�冷祭��� ���ᮡ�� ���䨣��権 $\leq 7$ ����� ����࠭�⢠
$\Bbb RP^3$, �� � ���� 楫�� �����饩 ࠡ���.

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\head
{\bf
2. ��㯮�冷祭�� ���ᮡ� ���䨣��樨 $\leq 7$ ����� ����࠭�⢠
$\Bbb RP^3$}
\endhead

����� $A$ --- ���ᮡ�� ���䨣���� ����� ����࠭�⢠ $\Bbb RP^3$.
���ᮡ�� ���䨣���� �����, ����祭��� �� �⮡ࠦ���� ᨬ���ਨ
���䨣��樨 $A$ �⭮�⥫쭮 �����ன ����௫�᪮�� � $\Bbb RP^3$,
���뢠���� {\it ��ઠ��� ��ࠧ��\/} ���䨣��樨 $A$. ����⭮, ��
���⪮-���⮯��᪨� ⨯ ��ઠ�쭮�� ��ࠧ� ��।������
���⪮-���\-�\-��\-�\-᪨� ⨯�� ��室��� ���䨣��樨 � �� ������ ��
�롮� ����௫�᪮��.

����� $B$, $C$ --- ���ᮡ� ���䨣��樨 ����� ����࠭�⢠
$\Bbb RP^3$. �㤥� �������, �� ���䨣���� $B$ {\it ��㡮
���⮯��\/} ���䨣��樨 $C$, �᫨ ���䨣���� $B$ ���⪮
���⮯�� ���� ���䨣��樨 $C$, ���� �� ��ઠ�쭮�� ��ࠧ�.

{\bf 2.1. �����} ��㯯� �����஬�� ��㡮 ���⮯���
��㯮�冷祭��� ���ᮡ�� ���䨣��権 ����� ����࠭�⢠
$\Bbb RP^3$ ��������.


\underbar{������⥫��⢮} �祢����.

����� $L^1$ � $L^2$ --- ��� �ਥ��஢���� �����ᥪ��騥��
���� �ਥ��஢������ ����࠭�⢠ $\Bbb RP^3$, �����樥��
��楯����� ������ ������⥫��. ����� $A_i^j$ --- ����୮ ࠧ����
�窨 ��אַ� $L^j$, $j=1,2$, $i=1,2,\,\dots,\,m$, ⠪��, �� �����⠭��
������ $i$ ᮣ��ᮢ��� � �ਥ��樥� ��אַ� $L^j$. ���ᬮ�ਬ
�������� ����⠭���� $\sigma$ �⥯��� $m$ � ᮥ����� ���묨
�窨 $A_i^1$ � $A_{\sigma (i)}^2$. �������� ��㯮�冷祭���
���ᮡ�� ���䨣���� $m$ ����� ����࠭�⢠ $\Bbb RP^3$, �����
������砥��� ᨬ����� $\op{jc}\(\sigma )$. ����� ������, ��
�⮡ࠦ����, ᮯ��⠢���饥 �����ன ����⠭���� $\sigma$
���⪮-���⮯��᪨� ⨯ ���䨣��樨 $\op{jc}\(\sigma),$
��।����� ���४⭮.

����� ��� ᮪�饭�� ����� �� �㤥� ��������� ����⠭����
$$
\sigma = \left (\matrix{ 1 & \dots & m \cr \sigma_1 & \dots & \sigma_m
} \right )
$$
ᨬ����� $(\sigma_1,\,\ldots,\,\sigma_m)$.

{\bf 2.2. ���६� \cite6} �� ��㯮�冷祭��� ���ᮡ��
���䨣���� $\leq 6$ ����� ����࠭�⢠ $\Bbb RP^3$ ��㡮
���⮯�� ����� �� ᫥����� ����୮ ��㡮 �����⮯��� ���䨣��権:

$\op{jc}\(1)$, $\op{jc}\(1,2)$, $\op{jc}\(1,2,3)$, $\op{jc}\(1,2,3,4)$,
$\op{jc}\(1,2,4,3)$, $\op{jc}\(1,2,3,4,5)$,
\break
$\op{jc}\(1,3,4,5,2)$, $\op{jc}\(1,4,5,2,3)$, $\op{jc}\(1,3,5,2,4)$,
$\op{jc}\(1,2,3,4,5,6)$,
\break
$\op{jc}\(1,2,3,4,6,5)$, $\op{jc}\(1,2,3,5,6,4)$, $\op{jc}\(1,2,4,3,6,5)$,
$\op{jc}\(1,2,4,6,3,5)$,
\break
$\op{jc}\(1,2,5,6,3,4)$, $\op{jc}\(1,2,3,6,5,4)$,
$\op{jc}\(1,3,5,2,6,4)$, $\op{jc}\(1,2,4,6,5,3)$,
\break
$L$, $M$.



\subhead
\indent
����砭��
\endsubhead
���ᠭ�� ���ᮡ�� ���䨣��権 $L$ � $M$ ����� ���� � \cite3 ���
\cite6.

{\bf 2.3. ���६� \cite5} �� ��㯮�冷祭��� ���ᮡ��
���䨣���� $7$ ����� ����࠭�⢠ $\Bbb RP^3$ ��㡮
���⮯�� ����� �� 37 ����୮ ��㡮 �����⮯��� ���䨣��権,
����᫥���� � �ਫ������ B �� \cite5.


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\head
{\bf
3. ��㯯� �����஬�� ��㯮�冷祭��� ���ᮡ�� ���䨣��権 $\leq 7$
����� ����࠭�⢠ $\Bbb RP^3$}
\endhead

���� ���� �ᯮ�짮������ ᫥���騥 ������祭��:

$\bold S_m$ --- ᨬ�����᪠� ��㯯� �⥯��� $m$;

${\Bbb Z}_m$ --- 横���᪠� ��㯯� ���浪� $m$;

$\bold D_m$ --- ����ࠫ쭠� ��㯯� ���浪� $2m$;

$\bold A_m$ --- �������६����� ��㯯� �⥯��� $m$.

{\bf 3.1. ���६�} ��㯯� �����஬�� ��㯮�冷祭��� ���ᮡ��
���䨣��権 $\leq 6$ ����� ����࠭�⢠ $\Bbb RP^3$ ����᫥��
� ᫥���饩 ⠡���:

$\op{jc}\(1)$ --- $\{1\}$;

$\op{jc}\(1,2)$ --- $\bold S_2$;

$\op{jc}\(1,2,3)$ --- $\bold S_3$;

$\op{jc}\(1,2,3,4)$ --- $\bold S_4$;

$\op{jc}\(1,2,4,3)$ --- ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

$\op{jc}\(1,2,3,4,5)$ --- $\bold S_5$;

$\op{jc}\(1,3,4,5,2)$ --- ${\bold S_2} \times {\bold S_3}$;

$\op{jc}\(1,4,5,2,3)$ --- ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

$\op{jc}\(1,3,5,2,4)$ --- $\bold D_5$;

$\op{jc}\(1,2,3,4,5,6)$ --- $\bold S_6$;

$\op{jc}\(1,2,3,4,6,5)$ --- ${\bold S_2} \times {\bold S_4}$;

$\op{jc}\(1,2,3,5,6,4)$ --- ${\bold S_2} \times {\bold S_3}$;

$\op{jc}\(1,2,4,3,6,5)$ --- ${{\Bbb Z}_2} \times {\bold D_4}$;

$\op{jc}\(1,2,4,6,3,5)$ --- ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

$\op{jc}\(1,2,5,6,3,4)$ --- $\langle a, b; \, a^2=1, b^6=1, {(ab)}^4=1,
ab^3=b^3a \rangle$ (����� $a$, $b$ --- ��ࠧ��騥 ��㯯� �����஬��,
� $a^2=1$, $b^6=1$, ${(ab)}^4=1$, $ab^3=b^3a$ --- �� ��।����騥
ᮮ⭮襭��);

$\op{jc}\(1,2,3,6,5,4)$ --- ${\bold S_3} \times {\bold S_3}$;

$\op{jc}\(1,3,5,2,6,4)$ --- ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

$\op{jc}\(1,2,4,6,5,3)$ --- ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

$L$ --- $\langle a, b; \, a^2=1, b^6=1, {(ab)}^4=1, ab^3=b^3a \rangle$;

$M$ --- $\bold A_5$.



{\bf 3.2. ���६�} ��㯯� �����஬�� ��㯮�冷祭��� ���ᮡ��
���䨣��権 $7$ ����� ����࠭�⢠ $\Bbb RP^3$ ����᫥�� �
᫥���饩 ⠡��� (�㬥��� ᮮ⢥����� �㬥�樨 �ਫ������ B
�� \cite5):

1) $\bold S_7$;

2) ${\bold S_2} \times {\bold S_5}$;

3) ${\bold S_2} \times {\bold S_4}$;

4) $\langle a, b; \, a^4=1, b^6=1, {(ba^2)}^2=1, {(b^3a)}^2=1 \rangle$;

5) $\langle a, b, c; \, a^{12}=1, b^2=1, c^2=1, ba=a^5b, ca=a^7c,
bc=cb \rangle$;

6) $\bold D_6$;

7) $\langle a, b, c; \, a^{12}=1, b^2=1, c^2=1, ba=a^5b, ca=a^7c,
bc=cb \rangle$;

8) ${{\Bbb Z}_2} \times {\bold D_4}$;

9) $\bold D_4$;

10) $\bold D_4$;

11) $\bold D_4$;

12) ${\bold S_2} \times {\bold S_2} \times {\bold S_2}$;

13) ${\bold S_3} \times {\bold S_4}$;

14) $\bold D_7$;

15) $\bold D_7$;

16) ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

17) $\bold D_6$;

18) $\bold D_{10}$;

19) $\bold D_5$;

20) $\bold D_{10}$;

21) ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

22) ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

23) $\langle a, b; \, a^2=1, b^6=1, {(ab)}^4=1, ab^3=b^3a \rangle$;

24) ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

25) ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

26) $\bold D_3$;

27) $\bold D_3$;

28) ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

29) $\bold D_4$;

30) $\bold D_6$;

31) $\bold D_6$;

32) $\bold D_6$;

33) ${{\Bbb Z}_2}$;

34) ${{\Bbb Z}_2} \times {\bold D_4}$;

35) ${{\Bbb Z}_2} \times {\bold D_4}$;

36) ${{\Bbb Z}_2} \times {{\Bbb Z}_2}$;

37) ${\bold S_2} \times {\bold S_2} \times {\bold S_3}$.



\underbar{������⥫��⢮ ⥮६ 3.1 � 3.2}.
�������� \cite{4} ��������� ��� �� ��㯮�冷祭��� ���ᮡ��
���䨣��樨 $A$ $m \leq 7$ ����� ����࠭�⢠ $\Bbb RP^3$ (��
�����஬ 䨪�஢����� 㯮�冷祭�� �� �����) ��।�����
������⢮ ������⮢ ��ࠧ� �������䨧�� ��㯯� $\op{Aut}\(A)$
� ᨬ�������� ��㯯� $\bold S_m$. ����� ��ࠧ��, ���
�।�⠢�⥫� ������� �� ��㡮-���⮯��᪨� ⨯�� ��㯮�冷祭���
���ᮡ�� ���䨣��権 $m \leq 7$ ����� ����࠭�⢠ $\Bbb RP^3$
�� ����� ��।����� �����஥ ���������⢮ ������⮢ ��㯯�
$\bold S_m$, �����饥 ��㯯� �����஬�� ������� �।�⠢�⥫�.
��᪮��� �� �⮬ ��������, �� �� ���ਢ����� (� �窨
�७�� �����䨪�樨 � ��� ������묨 �����묨 ��㯯���)
��㯯� �����஬�� ���ᮡ�� ���䨣��権 $\leq 7$ ����� ����࠭�⢠
$\Bbb RP^3$ ����� ������訥 ���浪�, � ��� ��� ����㤭� ��।�����
��������� ��⥬� ��ࠧ���� � ��।������ ᮮ⭮襭�� (�.,
����., \cite{2}). ��� ⮣� �⮡� �������஢��� ����祭��
���।�⠢����� ��㯯 �����஬�� � ���।�⠢����ﬨ ���
�������� ������� ��㯯, �ᯮ�짮������ ⠡���� �.~������
� �.~����� �� \cite{2}.

\subhead
\indent
����砭��
\endsubhead
���६� 3.1 �������� ����஢᪨� � �.�.~��䮭����, ⥮६� 3.2
�������� ����஢᪨� � �.�.~���㡥���.
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����� �ਧ��⥫�� �.~�.~��������᪮�� �� �� 業��� ����砭��.
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\centerline{\bf ���᮪ �ᯮ�짮������ ���p���p�}
\vskip .3 cm

\noindent
\hangindent 0.5cm
1. {\it ��� �.~�.} ���������᪨� ����� � ����� � �窠� ��嬥୮��
����࠭�⢠ // ��� ����. 1985. �.~284. $\No$~5. �.~1049 -- 1052.

\noindent
\hangindent 0.5cm
2. {\it ������ �., ����� �.} ��஦���騥 �������� � ��।����騥
ᮮ⭮襭�� ������� ��㯯. �., 1980. 240 �.

\noindent
\hangindent 0.5cm
3. {\it ����஢᪨� �.~�.} ���䨣��樨 ��� ��騢������ ����� //
���. ����. ᥬ���p�� ����. 1988. �.~167. �.~121 -- 134.

\noindent
\hangindent 0.5cm
4. {\it ����஢᪨� �.~�., ������ �.~�.} �����䨪��� 㯮�冷祭���
���ᮡ�� ���䨣��権 $\leq 7$ ����� ����࠭�⢠ $\Bbb RP^3$
� �筮���� �� ���⪨� ���⮯�� // ��� ��. 1995.
�.~231. �.~269 -- 285.

\noindent
\hangindent 0.5cm
5. {\it Borobia A., Mazurovskii V.~F.} Nonsingular configurations of $7$
lines of $\Bbb RP^3$ // J. Knot Theory and Ramifications. 1997.
Vol.~6. $\No$~6. P.~751 -- 783.

\noindent
\hangindent 0.5cm
6. {\it Mazurovskii V.~F.} Configurations of at most $6$ lines of
$\Bbb RP^3$ // Lect. Notes in Math. 1992. Vol.~1524. P.~354---371.

\end
