grupuri de simetrii
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5/23/2018 Grupuri de Simetrii
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S
S
( , )
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S
S
( , )
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P
: P P R
( , ) , , P ( , ) = =
(
,
) =
(
,
)
( , ) ( , ) + ( , ), , , P ( , ) = ( , ) + ( , )
P
: P P
( ( ), ( )) = ( , ), , P.
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P
: P P R
( , ) , , P ( , ) = =
(
,
) =
(
,
)
( , ) ( , ) + ( , ), , , P ( , ) = ( , ) + ( , )
P
: P P
( ( ), ( )) = ( , ), , P.
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O( )
O( ) =
M
(R) | = =
=
( ,R) | =
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O( )
O( ) =
M
(R) | = =
=
( ,R) | =
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O( )
O( ) =
M
(R) | = =
=
( ,R) | =
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F
F P P F : P P F
(F) = F
F P P
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F
F P P F : P P F
(F) = F
F P P
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= { ,
,
,
}
= <
,
= , >
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D
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D
, , , , , , , ,
=
, ,
D
=
, , , = , , , ,
= =
= = = =
= =
=
=
=
D
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D
, , , , , , , ,
=
, ,
D
=
, , , = , , , ,
= =
= = = =
= =
=
=
=
D
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D
, , , , , , , ,
=
, ,
D
=
, , , = , , , ,
= =
= = = =
= =
=
=
=
D
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D
, ,
= () = = ,
= ( ) = = ,
= (
)
=
=.
D
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D
C =
, ,
,
=< >
D
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D
C =
, ,
,
=< >
C
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C
D
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D
=
= ,
= =
D
=
, , , , ,
= () = =
= ( ) = =
D
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D
C
=
, ,
C
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C
D
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D
=
, , , , , , , , , ,
, = ,
C
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C
= { , , , , }
D C
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, , , , , , , ,
D
=< , >
C
=< >
D
C
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, , , , , , , ,
D
=< , >
C
=< >
D
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= =
(
)
=
,
,
= , , .
D
=< > C
= { }
D
=< , , = >= C = { , ,}
D
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= =
(
)
=
,
,
= , , .
D
=< > C
= { }
D
=< , , = >= C = { , ,}
D
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= =
(
)
=
,
,
= , , .
D
=< > C
= { }
D
=< , , = >= C = { , ,}
D
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= =
(
)
=
,
,
= , , .
D
=< > C
= { }
D
=< , , = >= C = { , ,}
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N D
C
D
C
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N D
C
D
C
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N D
C
D
C
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C
D
D
C
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C
D
D
C
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C
D
G P
G
G
G= C
= { }
, G, , =
, G =
,
, , , G
G
N : = G = G
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N : , = , G
, = , G
,
G
, G, N
=
,
= ,
,
G=< ,
>= C
, ,
=
G
G
C
= { , , , } G
N : = G = G
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N : , = , G
, = , G
,
G
, G, N
=
,
= ,
,
G=< ,
>= C
, ,
=
G
G
C
= { , , , } G
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G ,, , ,
=
G=
G=
, , , ,,, , ,
=
G=< >= D
>
, , G= D
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G ,, , ,
=
G=
G=
, , , ,,, , ,
=
G=< >= D
>
, , G= D
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