A Study on p-Maps through Right Transversals
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Date
2015
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Sam Higginbottom Institute of Agriculture, Technology & Sciences (SHIATS)
Abstract
The thesis has been divided into five chapters. The first chapter is the introduction of
the whole thesis. The second chapter is devoted to review of the literature in which we
have tried to show the researches in group theory.
In the third chapter namely, methodology, we have defined concepts which are
necessary tools to make the thesis self contained. This is divided into four sections.
First section contain information about loops, c-groupoid and c-homomorphism. In
second section, we have talked about some general topology. Third section is devoted
to categories and functors. In fourth section we have introduced homotopy.
Fourth chapter namely, result and discussion, is the main chapter of the thesis. This
chapter is divided into four sections.
First section of it is devoted to the study of p-maps and results based on them. The
following are some significant results of this part:
Proposition (4.1.2.6): LetG be a group with identity e and p :GG be a pmap .
The subset p(G) p(g) : gG of G is a subgroup of G .
Proposition (4.1.2.9): Let G be a group with identity e and p :GG be a pmap .
Let S be a subset defined in proposition (4.1.2.7). Define a binary operation on S
by -1 x y p(xy) xy for all x, yS . Then (S, ) is a right loop.
Proposition (4.1.2.13): Let G be a group with identity e and p :GG be a
pmap . Let S be a right transversal to p(G) in G . Let us define : S p(G)S by
1 x h p(xh) xh . Then is a right action of p(G) on S .
Proposition (4.1.3.1): Let G be a group with identity e and p :GG be a pmap .
Let S be a right transversal with identity to p(G) in G . If pmap satisfies the
condition 1 2 1 2 p(g g ) p(g p(g )) , then the right loop (S, ) is a group.
ii
Proposition (4.1.3.11): Let G be a group with identity e . Then the total number of
distinct pmaps in G is the total number of distinct factorizations of G as HS where
H is a subgroup of G and S is a right transversal (with identity e ) of H in G .
Theorem (4.1.3.15): Let G be a group with identity e and p be a pmap . Then G
be an extension of the subgroup p(G) with a right transversal S to p(G) in G .
In section second of it, we have described another map, namely p-tilda map and then
proved some results:
Proposition (4.2.2.6): Let G be a group with identity e and p :GG be a p -map .
Then the set H {gG: p(g) e} is a subgroup of G .
Proposition (4.2.2.9) [67]: Let G be a group with identity e and p :GG be a
p -map . Let 1 2 g , g G . Let S be a subset defined in proposition (4.2.2.7) . Define a
binary operation on S by 1 2 1 2 p(g ) p(g ) p(p(g )p(g )) for all 1 2 p(g ),p(g )S . Then
(S, ) is a right loop.
Proposition (4.2.3.1): Let G be a group with identity e and p :GG be a
p -map . Let S be a right transversal with identity to H in G . If p -map satisfies the
condition 1 2 1 2 p(g p(g )) p(g )p(g ) for all 1 2 g , g G , then the right loop (S, ) is a
group.
Theorem (4.2.4.3): Let G be a group with identity e and p :GG be a p -map .
Let H and S be as defined in proposition (4.2.2.6) and (4.2.2.7) respectively. Then
G be an extension of the subgroup H with a right transversal S to H in G .
In section third of it, we have introduced the concept of H-group and H-transversal
and proved the following important results:
Proposition (4.3.2.14): Let 0 (X, x ) and 0 (Y, y ) be two pointed topological spaces.
Then 0 X { : : I X is a loop based at x } is an H-group with continuous
multiplication . Similarly 0 Y { : : I Y is a loop based at y } is an Hiii
group with continuous multiplication . Let f : (Y, y0 )(X, x0 ) is a continuous map.
Then (Y,) is an H-subgroup together with an H-map ( , ) ( , ) f Y X .
Theorem (4.3.3.2): Let (G,) be an H-group with base point identity element e of
the group G . Let p be an H-transversal in an H-group (G,) . Then there is a
canonical H-group structure on p(G) with respect to which the inclusion ( ) i p G G
is an H-subgroup of (G,) .
In section four of it, we have introduced another H-transversal for an H-group and
proved the following important result:
Theorem (4.4.2.2): Let (G,) be an H-group with base point identity element e of
the group G . Let p be an H-transversal in an H-group (G,) . Then p(G) is an Hgroup
with respect to the operation defined as follows
1 2 1 2 ( p(g ), p(g )) ( p )( p(g ), p(g )) for all 1 2 g , g G .
In the chapter fifth, namely summary and conclusion, we have summaries the whole
thesis, by mentioning that the following four are the main results of the thesis.
Theorem (4.1.3.15): Let G be a group with identity e and p be a pmap . Then G
be an extension of the subgroup p(G) with a right transversal S to p(G) in G .
Theorem (4.2.4.3): Let G be a group with identity e and p :GG be a p -map .
Let H and S be as defined in proposition (4.2.2.6) and (4.2.2.7) respectively. Then
G be an extension of the subgroup H with a right transversal S to H in G .
Theorem (4.3.3.2): Let (G,) be an H-group with base point identity element e of
the group G . Let p be an H-transversal in an H-group (G,) . Then there is a
canonical H-group structure on p(G) with respect to which the inclusion ( ) i p G G
is an H-subgroup of (G,) .
Theorem (4.4.2.2): Let (G,) be an H-group with base point identity element e of
the group G . Let p be an H-transversal in an H-group (G,) . Then p(G) is an Hiv
group with respect to the operation defined as follows
1 ( p(g ), p(g2)) ( p )( p(g1), p(g2)) for all 1 2 g , g G .
In the end of chapter five, we have shown some importance and application of group
theory in different areas.
Hence, we have worked upto some extent in the direction of the existing problem of
classifying all finite groups by determining all groups G (up to isomorphism) with a
fixed subgroup H as a normal subgroup such that the quotient group G/ H is also a
given group K . By making p and p to be continuous, we have also worked on Hgroup
and H-transversal.
Description
Thesis titled “A Study on p-Maps through Right Transversals’’” submitted in partial fulfillment of the requirements for the award of the degree of Doctor of Philosophy in Mathematics by Punish Kumar.