Is there a general way to simplify such group presentations (Free Abelian Group with Relations)?
$begingroup$
If I want to simplify the group presentation (free abelian group with relations)
$$langle a,b,cmid 2a=b=2crangle,$$
I can simplify it as $$langle a,cmid 2a=2crangleconglangle a,a-cmid 2(a-c)ranglecongmathbb{Z}oplusmathbb{Z}_2.$$
How about for more complicated presentations such as
$$langle a,b,cmid 2a=3b=5crangle?$$
How do I simplify it, if possible?
In general, I am interested in simplifying
$$langle a,b,cmid n_1a=n_2b=n_3crangle,$$
where $n_1,n_2,n_3inmathbb{Z}$.
Thanks.
abstract-algebra group-theory group-presentation combinatorial-group-theory
edited Dec 3 '18 at 0:43
Shaun
8,883113681
asked Feb 5 '18 at 7:41
yoyosteinyoyostein
7,91293768
$endgroup$
add a comment |
$begingroup$
If I want to simplify the group presentation (free abelian group with relations)
$$langle a,b,cmid 2a=b=2crangle,$$
I can simplify it as $$langle a,cmid 2a=2crangleconglangle a,a-cmid 2(a-c)ranglecongmathbb{Z}oplusmathbb{Z}_2.$$
How about for more complicated presentations such as
$$langle a,b,cmid 2a=3b=5crangle?$$
How do I simplify it, if possible?
In general, I am interested in simplifying
$$langle a,b,cmid n_1a=n_2b=n_3crangle,$$
where $n_1,n_2,n_3inmathbb{Z}$.
Thanks.
abstract-algebra group-theory group-presentation combinatorial-group-theory
edited Dec 3 '18 at 0:43
Shaun
8,883113681
asked Feb 5 '18 at 7:41
yoyosteinyoyostein
7,91293768
$endgroup$
2
$begingroup$
The general method for this problem is the Smith Normal Form diagonlization algorithm for matrices over the integers.
$endgroup$
– Derek Holt
Feb 5 '18 at 9:46
add a comment |
$begingroup$
If I want to simplify the group presentation (free abelian group with relations)
$$langle a,b,cmid 2a=b=2crangle,$$
I can simplify it as $$langle a,cmid 2a=2crangleconglangle a,a-cmid 2(a-c)ranglecongmathbb{Z}oplusmathbb{Z}_2.$$
How about for more complicated presentations such as
$$langle a,b,cmid 2a=3b=5crangle?$$
How do I simplify it, if possible?
In general, I am interested in simplifying
$$langle a,b,cmid n_1a=n_2b=n_3crangle,$$
where $n_1,n_2,n_3inmathbb{Z}$.
Thanks.
abstract-algebra group-theory group-presentation combinatorial-group-theory
edited Dec 3 '18 at 0:43
Shaun
8,883113681
asked Feb 5 '18 at 7:41
yoyosteinyoyostein
7,91293768
$endgroup$
If I want to simplify the group presentation (free abelian group with relations)
$$langle a,b,cmid 2a=b=2crangle,$$
I can simplify it as $$langle a,cmid 2a=2crangleconglangle a,a-cmid 2(a-c)ranglecongmathbb{Z}oplusmathbb{Z}_2.$$
How about for more complicated presentations such as
$$langle a,b,cmid 2a=3b=5crangle?$$
How do I simplify it, if possible?
In general, I am interested in simplifying
$$langle a,b,cmid n_1a=n_2b=n_3crangle,$$
where $n_1,n_2,n_3inmathbb{Z}$.
Thanks.
abstract-algebra group-theory group-presentation combinatorial-group-theory
abstract-algebra group-theory group-presentation combinatorial-group-theory
edited Dec 3 '18 at 0:43
Shaun
8,883113681
asked Feb 5 '18 at 7:41
yoyosteinyoyostein
7,91293768
edited Dec 3 '18 at 0:43
Shaun
8,883113681
asked Feb 5 '18 at 7:41
yoyosteinyoyostein
7,91293768
edited Dec 3 '18 at 0:43
Shaun
8,883113681
edited Dec 3 '18 at 0:43
Shaun
8,883113681
edited Dec 3 '18 at 0:43
Shaun
8,883113681
8,883113681
asked Feb 5 '18 at 7:41
yoyosteinyoyostein
7,91293768
asked Feb 5 '18 at 7:41
yoyosteinyoyostein
7,91293768
asked Feb 5 '18 at 7:41
yoyosteinyoyostein
7,91293768
7,91293768
2
$begingroup$
The general method for this problem is the Smith Normal Form diagonlization algorithm for matrices over the integers.
$endgroup$
– Derek Holt
Feb 5 '18 at 9:46
add a comment |
2
$begingroup$
The general method for this problem is the Smith Normal Form diagonlization algorithm for matrices over the integers.
$endgroup$
– Derek Holt
Feb 5 '18 at 9:46
2
2
$begingroup$
The general method for this problem is the Smith Normal Form diagonlization algorithm for matrices over the integers.
$endgroup$
– Derek Holt
Feb 5 '18 at 9:46
$begingroup$
The general method for this problem is the Smith Normal Form diagonlization algorithm for matrices over the integers.
$endgroup$
– Derek Holt
Feb 5 '18 at 9:46
add a comment |
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$begingroup$
The general method for this problem is the Smith Normal Form diagonlization algorithm for matrices over the integers.
$endgroup$
– Derek Holt
Feb 5 '18 at 9:46