#9773: Abelian groups
---------------------------+------------------------------------------------
Reporter: rbeezer | Owner: AlexGhitza
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-4.5.3
Component: algebra | Keywords:
Author: Rob Beezer | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
---------------------------+------------------------------------------------
Changes (by newvalueoldvalue):
* cc: davidloeffler, cremona, was, nthiery, boothby, jason, kcrisman
(added)
* status: new => needs_info
* type: defect => enhancement
* author: => Rob Beezer
Comment:
AAG is the class of additive abelian groups. This is an infinite group
with a subgroup and a quotient. (Typically quotients lose the generators
and are "generic" but not in this example.)
{{{
sage: A=AAG([3,4,0])
sage: A.gens()
((2, 3, 0), (0, 0, 1))
sage: A.0.order()
12
sage: A.1.order()
+Infinity
sage: B=A.subgroup([A.1])
sage: B
Infinite additive abelian group isomorphic to Z with generator(s): (0, 0,
1)
sage: C=A/B
sage: C
Finite additive abelian group isomorphic to Z_12 with generator(s): (2, 3,
0)
}}}
GUN is a constructor of Groups of Units Mod n. It employs MAG, the class
of multiplicative abelian groups. This is an intersection of two
subgroups, and then a Cayley table is free (in the category of
multiplicative groups).
{{{
sage: G=GUN(72)
sage: G.list()
[1, 65, 49, 17, 25, 41, 55, 47, 31, 71, 7, 23, 37, 29, 13, 53, 61, 5, 19,
11, 67, 35, 43, 59]
sage: G.subgroup([71,7])
Finite multiplicative abelian group isomorphic to Z_2 + Z_6 with
generator(s): 55, 65
sage: K=G.subgroup([71,7])
sage: K.list()
[1, 65, 49, 17, 25, 41, 55, 47, 31, 71, 7, 23]
sage: L=G.subgroup([13,7])
sage: L
Finite multiplicative abelian group isomorphic to Z_2 + Z_6 with
generator(s): 55, 61
sage: L.list()
[1, 61, 49, 37, 25, 13, 55, 43, 31, 19, 7, 67]
sage: M=K.intersection(L)
sage: M.list()
[1, 7, 49, 55, 25, 31]
sage: M
Finite multiplicative abelian group isomorphic to Z_6 with generator(s): 7
sage: M.cayley_table()
* a b c d e f
+------------
a| a b c d e f
b| b c d e f a
c| c d e f a b
d| d e f a b c
e| e f a b c d
f| f a b c d e
}}}
This is an example from the current additive abelian wrapper class. It
shows the generators keyword allowing arbitrary elements to form the
group, so long as they know how to add. GUN above is similar, but with
multiplication.
{{{
sage: E = EllipticCurve('30a2')
sage: pts = [E(4,-7,1), E(7/4, -11/8, 1), E(3, -2, 1)]
sage: M=AAG([3,2,2], generators=pts)
sage: M.list()
[(0 : 1 : 0), (13 : -52 : 1), (4 : -7 : 1), (3 : -2 : 1), (4 : 2 : 1), (13
: 38 : 1), (7/4 : -11/8 : 1), (1 : -4 : 1), (-2 : -7 : 1), (-5 : 2 : 1),
(-2 : 8 : 1), (1 : 2 : 1)]
sage: M.gens()
((7/4 : -11/8 : 1), (13 : -52 : 1))
sage: N=M.subgroup([M.1])
sage: N
Finite additive abelian group isomorphic to Z_6 with generator(s): (13 :
-52 : 1)
sage: N.list()
[(0 : 1 : 0), (13 : -52 : 1), (4 : -7 : 1), (3 : -2 : 1), (4 : 2 : 1), (13
: 38 : 1)]
}}}
There is lots to do here still: different filenames, different class
names, error-checking, doctests, comparisons, and so on. But the code
seems to be working. I'm not 100% confident on the {{{__call__}}} method
of the main abstract class and I don't know if I need some things to
support coercion better. Any advice or comments at this stage would be
appreciated before I begin to clean this all up.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9773#comment:1>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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