Register to reply

Finding a set of Generators for a group G when Generators for Kerh, Imh are known; h

by Bacle
Tags: generators, kerh
Share this thread:
Bacle
#1
May8-11, 04:47 PM
P: 662
Hi, Algebraists:

Say h:G-->G' is a homomorphism between groups, and that we know a set
of generators {ki} for Imh:=h(G)<G' , and we also know of a set of generators
{b_j} for Kerh . Can we use these two sets {ki} and {bj} of generators for
Imh and Kerh respectively, to produce a set of generators for G itself?

It looks a bit like the group extension problem (which I know very little about,
unfortunately).

This is what I have tried so far :


We get a Short Exact Sequence:

1 -->Kerh -->G-->Imh -->1

But I am not sure this sequence necessarily splits (if it doesn't split, then you must acquit!)



It would seem like we could pull-back generators of Imh back into G, i.e., for any g in G, we can write h(g)=Product{$k_i$ $e_i$} of generators in h(G).

Similarly, we know that G/Kerh is Isomorphic to h(G) , and that g~g' iff h(g)=h(g') ( so that,the isomorphism h':G/Kerh-->h(G) is given by h'([g]):=h(g) )

But I get kind of lost around here.

Any Ideas?

Thanks.
Phys.Org News Partner Science news on Phys.org
Security CTO to detail Android Fake ID flaw at Black Hat
Huge waves measured for first time in Arctic Ocean
Mysterious molecules in space
micromass
#2
May9-11, 09:49 AM
Mentor
micromass's Avatar
P: 18,061
My thoughts are this:

Take [tex]\{a_1,...,a_n\}[/tex] generators of ker(f), and take [tex]\{b_1,...,b_m\}[/tex]generators of im(f). For every bi, we can find a ci such that [tex]f(c_i)=b_i[/tex]. Then [tex]\{a_1,...,a_n,b_1,...,b_m\}[/tex] is a generating set for G.

Indeed, take g in G, then we can write f(g) as

[tex]f(g)=b_{i_1}...b_{i_j}=f(c_{i_1})...f(c_{i_j})=f(c_{i_1}...c_{i_j})[/tex]

Thus

[tex]f(gc_{i_1}^{-1}...c_{i_j}^{-1})\in Ker(f)[/tex],

so we can write

[tex]gc_{i_1}^{-1}...c_{i_j}^{-1}=a_{k_1}...a_{k_l}[/tex],

so that follows

[tex]g=c_{i_1}...c_{i_j}a_{k_1}...a_{k_l}[/tex]

We have writte g as a combination of the suitable elements, so the set [tex]\{a_1,...,a_n,c_1,...,c_m\}[/tex] is generating...


Register to reply

Related Discussions
Number of generators of SU(n) group High Energy, Nuclear, Particle Physics 3
Six Generators of Group SO(4) Quantum Physics 8
Number of generators of finite group Linear & Abstract Algebra 4
Cliffor group generators - matrix form ? Precalculus Mathematics Homework 0
Group Generators Calculus & Beyond Homework 6