3 Spacing layout had flaws; just removed extra-spacing.

Hi, Everyone:

I would appreciate some references for the version of Reidemeister-Schreier that is used to find the stabilizer of a point under a group action. The only refs. I have found are about Schreier-Sims method, but I have not been able to find anything on it.

The version of R-S I know of allows us to find a presentation of a subgroup H of a group G, by using transversals, etc.

I think there is a connection between the two, but I am not sure.

Sorry, I forgot to ask something important: I would like to know how the following process --the adaptation of R-S to group actions ( or maybe a version of Cayley graphs) produces a set of generators for the stabilizer of a fixed element sk , under a group action:

We start with a group action HxS-->S (could also be a left action), and we are given the (finite) set {$h_1$,..,$h_n$} of generators for H; S is a finite set. We then define a graph G by:

1)The vertices are the elements sj of S

2)We join $s_i$ with $s_k$ with an edge labeled $h_j$ , if $h_j$.$s_i$=$s_k$ , i.e., if the action of $h_j$ on $s_i$ results in sk.

3) We construct a spanning-tree T for G, rooted at $s_k$ (the element of S being stabilized); I think it is clear that G is connected --|n|-connected, actually, where n is the size of the generating set for H (tho we mayhave loops) , to guarantee the existence of a spanning- tree.

Claim: the edges in G-T generate the stabilizer Stb{{$s_k$})of $s_k$ under this action.

Anyone have a suggestion for showing this?

I don't remember the place where I read this, but I remember some related results:

The background/context is a generalization of the fact that , given a group H and any subgroup

H' of H , there is an action by H for which H' is the stabilizer. Specifically, this

action is the "standard" action of H on H/H' (standard group quotient); we just define, for

any h1H' on H/H' and h in H:

h.(h1H' ) --> (h.h1)H'

Then H' is the stabilizer of the coset eH'=H' .

I think this is also related to the method for finding the fundamental group of a rooted

connected graph G: we find a spanning-tree T. Then each edge e=(gi,gj) in G-T defines a non-

trivial element of $Pi_1$(G): we start at , say, $g_i$ (which is in T, since T spans) , then we

find the (unique; any two paths would form a loop in T) path $P_i$ in T from $g_i$ to the root g,

and from g we find the unique path $P_j$ to $g_j$; the other vertex in e. Then the composition

$P_i$$P_je forms a non-trivial loop in G. It is just a little more work to show that these edges freely generate the fundamental group. These are the results that were related to the issue of the stabilizer. Thanks for any Suggestions, Refs. 2 I added a second question I had forgotten to ask. stabilizer of a point under a group action. The only refs. I have found Thanks in Advance. Sorry, I forgot to ask something important: I would like to know how the following process --the adaptation of R-S to group actions ( or maybe a version of Cayley graphs) produces a set of generators for the stabilizer of a fixed element sk , under a group action: We start with a group action HxS-->S (could also be a left action), and we are given the (finite) set {h_1,..,h_n} of generators for H; S is a finite set. We then define a graph G by: 1)The vertices are the elements sj of S 2)We join s_i with s_k with an edge labeled h_j , if h_j.s_i=s_k , i.e., if the action of h_j on s_i results in sk.3) We construct a spanning-tree T for G, rooted at s_k (the element of S being stabilized); I think it is clear that G is connected --|n|-connected, actually, where n is the size of the generating set for H (tho we mayhave loops) , to guarantee the existence of a spanning- tree. Claim: the edges in G-T generate the stabilizer Stb{{s_k})of s_k under this action.Anyone have a suggestion for showing this? I don't remember the place where I read this, but I remember some related results:The background/context is a generalization of the fact that , given a group H and any subgroupH' of H , there is an action by H for which H' is the stabilizer. Specifically, thisaction is the "standard" action of H on H/H' (standard group quotient); we just define, forany h1H' on H/H' and h in H:h.(h1H' ) --> (h.h1)H'Then H' is the stabilizer of the coset eH'=H' . I think this is also related to the method for finding the fundamental group of a rootedconnected graph G: we find a spanning-tree T. Then each edge e=(gi,gj) in G-T defines a non-trivial element of Pi_1(G): we start at , say, g_i (which is in T, since T spans) , then wefind the (unique; any two paths would form a loop in T) path P_i in T from g_i to the root g,and from g we find the unique path P_j to g_j; the other vertex in e. Then the compositionP_i$$P_j$e  forms a non-trivial loop in G. It is just a little more work to show that theseedges freely generate the fundamental group.These are the results that were related to the issue of the stabilizer.Thanks for any Suggestions, Refs.

 
 
 
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# Reidemeister-Schreier Method for Finding Stabilizer of an Element in a Group Action

Hi, Everyone:

I would appreciate some references for the version of Reidemeister-Schreier that is used to find the Stabilizer of a point under a group action. The only refs. I have found are about Schreier-Sims method, but I have not been able to find anything on it.

The version of R-S I know of allows us to find a presentation of a subgroup H of a group G, by using transversals, etc.

I think there is a connection between the two, but I am not sure.