Ask Your Question
1

Direct product of $S_n$ and $\mathbb Z_m$

asked 2016-11-14 10:22:01 -0500

huhahu gravatar image

updated 2016-11-14 10:25:43 -0500

How do I generate cayley table for direct product of symmetric group $S_n$ and additive group of integers modulo $\mathbb Z/m\mathbb Z$ or $\mathbb Z_m$.

edit retag flag offensive close merge delete

1 answer

Sort by ยป oldest newest most voted
2

answered 2016-11-14 14:05:51 -0500

updated 2016-11-15 02:26:15 -0500

Use the cartesian_product and multiplication_table methods.

Is this what you are looking for?

sage: version()
'SageMath version 7.4, Release Date: 2016-10-18'

sage: G = SymmetricGroup(3).cartesian_product(CyclicPermutationGroup(3))
sage: G in Groups()
True
sage: G.multiplication_table()
*  a b c d e f g h i j k l m n o p q r
 +------------------------------------
a| a b c d e f g h i j k l m n o p q r
b| b c a e f d h i g k l j n o m q r p
c| c a b f d e i g h l j k o m n r p q
d| d e f a b c p q r m n o j k l g h i
e| e f d b c a q r p n o m k l j h i g
f| f d e c a b r p q o m n l j k i g h
g| g h i m n o j k l a b c p q r d e f
h| h i g n o m k l j b c a q r p e f d
i| i g h o m n l j k c a b r p q f d e
j| j k l p q r a b c g h i d e f m n o
k| k l j q r p b c a h i g e f d n o m
l| l j k r p q c a b i g h f d e o m n
m| m n o g h i d e f p q r a b c j k l
n| n o m h i g e f d q r p b c a k l j
o| o m n i g h f d e r p q c a b l j k
p| p q r j k l m n o d e f g h i a b c
q| q r p k l j n o m e f d h i g b c a
r| r p q l j k o m n f d e i g h c a b

To figure out who are a, b, c, etc, list the elements of the group.

sage: G.list()
[((), ()),
 ((), (1,2,3)),
 ((), (1,3,2)),
 ((1,2), ()),
 ((1,2), (1,2,3)),
 ((1,2), (1,3,2)),
 ((1,2,3), ()),
 ((1,2,3), (1,2,3)),
 ((1,2,3), (1,3,2)),
 ((1,3,2), ()),
 ((1,3,2), (1,2,3)),
 ((1,3,2), (1,3,2)),
 ((2,3), ()),
 ((2,3), (1,2,3)),
 ((2,3), (1,3,2)),
 ((1,3), ()),
 ((1,3), (1,2,3)),
 ((1,3), (1,3,2))]

The multiplication table with elements is hard to read.

    sage: G.multiplication_table(names ...
(more)
edit flag offensive delete link more

Comments

can't we have notation like $\mathbb Z_{3} = {\bar 0, \bar 1, \bar 2}$ and $S_{3}={ (), (1,2), (1,3), (2,3),(1,2,3), (1, 3, 2)}$ and $\mathbb Z_3 \times S_3 = { (\bar 0, ()), (\bar 0, (1,2)), \dots }$?? The one we encounter in

huhahu gravatar imagehuhahu ( 2016-11-14 21:58:02 -0500 )edit

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower

Stats

Asked: 2016-11-14 10:22:01 -0500

Seen: 50 times

Last updated: Nov 15 '16