Engel conditions in certain groups / Quek Shio Gai
This thesis is a study of certain Engel conditions. First, we will define the set of all the X-relative left Engel elements L(G;X) and the set of all the bounded X-relative left Engel elements L(G;X), where X is a subset of G. When X = G, L(G;X) = L(G) and L(G;X) = L(G), where L(G) is the set of...
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| Format: | Thesis |
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2015
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| Online Access: | http://studentsrepo.um.edu.my/6499/ http://studentsrepo.um.edu.my/6499/1/CD_title.pdf http://studentsrepo.um.edu.my/6499/2/thesis_3_v9.pdf http://studentsrepo.um.edu.my/6499/3/title4a_Front_cover_and_first_page.pdf http://studentsrepo.um.edu.my/6499/4/title4b_Side.pdf |
| _version_ | 1848773181949607936 |
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| author | Quek, Shio Gai |
| author_facet | Quek, Shio Gai |
| author_sort | Quek, Shio Gai |
| building | UM Research Repository |
| collection | Online Access |
| description | This thesis is a study of certain Engel conditions. First, we will define the set
of all the X-relative left Engel elements L(G;X) and the set of all the bounded
X-relative left Engel elements L(G;X), where X is a subset of G. When X = G,
L(G;X) = L(G) and L(G;X) = L(G), where L(G) is the set of all the usual left
Engel elements and L(G) is the set of all the usual bounded left Engel elements.
Next, we de�ne the X-relative Hirsch-Plotkin radical HP(G;X) and the X-relative
Baer radical B(G;X). When X = G, HP(G;X) = HP(G) and B(G;X) = B(G)
where HP(G) is the usual Hirsch-Plotkin radical and B(G) is the usual Baer radical.
We will show that if X is a normal solvable subgroup of G, then B(G;X) = L(G;X)
and HP(G;X) = L(G;X). This is an extension of the classical results B(G) = L(G)
and HP(G) = L(G) provided that G is solvable. Next, we show that if X is
a normal subgroup of G and G satis�es the maximal condition, then L(G;X) =
HP(G;X) = B(G;X) = L(G;X), which is also an extension of the classical result
L(G) = HP(G) = B(G) = L(G). We also proved similar results when X is a
subgroup of certain linear groups.
Let G be a group and h; g 2 G. The 2-tuple (h; g) is said to be an n-Engel pair,
n � 2, if h = [h;n g]; g = [g;n h] and h 6= 1. We will show that the subgroup generated
by the 5-Engel pair (x; y) satisfying yxy = xyx and x5 = 1 is the alternating
group A5. Next, we show that if (x; y) is an n-Engel pair, xyx |
| first_indexed | 2025-11-14T13:38:20Z |
| format | Thesis |
| id | um-6499 |
| institution | University Malaya |
| institution_category | Local University |
| last_indexed | 2025-11-14T13:38:20Z |
| publishDate | 2015 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | um-64992016-09-27T05:03:19Z Engel conditions in certain groups / Quek Shio Gai Quek, Shio Gai Q Science (General) This thesis is a study of certain Engel conditions. First, we will define the set of all the X-relative left Engel elements L(G;X) and the set of all the bounded X-relative left Engel elements L(G;X), where X is a subset of G. When X = G, L(G;X) = L(G) and L(G;X) = L(G), where L(G) is the set of all the usual left Engel elements and L(G) is the set of all the usual bounded left Engel elements. Next, we de�ne the X-relative Hirsch-Plotkin radical HP(G;X) and the X-relative Baer radical B(G;X). When X = G, HP(G;X) = HP(G) and B(G;X) = B(G) where HP(G) is the usual Hirsch-Plotkin radical and B(G) is the usual Baer radical. We will show that if X is a normal solvable subgroup of G, then B(G;X) = L(G;X) and HP(G;X) = L(G;X). This is an extension of the classical results B(G) = L(G) and HP(G) = L(G) provided that G is solvable. Next, we show that if X is a normal subgroup of G and G satis�es the maximal condition, then L(G;X) = HP(G;X) = B(G;X) = L(G;X), which is also an extension of the classical result L(G) = HP(G) = B(G) = L(G). We also proved similar results when X is a subgroup of certain linear groups. Let G be a group and h; g 2 G. The 2-tuple (h; g) is said to be an n-Engel pair, n � 2, if h = [h;n g]; g = [g;n h] and h 6= 1. We will show that the subgroup generated by the 5-Engel pair (x; y) satisfying yxy = xyx and x5 = 1 is the alternating group A5. Next, we show that if (x; y) is an n-Engel pair, xyx 2015 Thesis NonPeerReviewed application/pdf http://studentsrepo.um.edu.my/6499/1/CD_title.pdf application/pdf http://studentsrepo.um.edu.my/6499/2/thesis_3_v9.pdf application/pdf http://studentsrepo.um.edu.my/6499/3/title4a_Front_cover_and_first_page.pdf application/pdf http://studentsrepo.um.edu.my/6499/4/title4b_Side.pdf Quek, Shio Gai (2015) Engel conditions in certain groups / Quek Shio Gai. PhD thesis, University of Malaya. http://studentsrepo.um.edu.my/6499/ |
| spellingShingle | Q Science (General) Quek, Shio Gai Engel conditions in certain groups / Quek Shio Gai |
| title | Engel conditions in certain groups / Quek Shio Gai
|
| title_full | Engel conditions in certain groups / Quek Shio Gai
|
| title_fullStr | Engel conditions in certain groups / Quek Shio Gai
|
| title_full_unstemmed | Engel conditions in certain groups / Quek Shio Gai
|
| title_short | Engel conditions in certain groups / Quek Shio Gai
|
| title_sort | engel conditions in certain groups / quek shio gai |
| topic | Q Science (General) |
| url | http://studentsrepo.um.edu.my/6499/ http://studentsrepo.um.edu.my/6499/1/CD_title.pdf http://studentsrepo.um.edu.my/6499/2/thesis_3_v9.pdf http://studentsrepo.um.edu.my/6499/3/title4a_Front_cover_and_first_page.pdf http://studentsrepo.um.edu.my/6499/4/title4b_Side.pdf |