Product decompositions of quasirandom groups and a Jordan type theorem
We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 = G. We use this to obtain improved versions of recen...
| Auteurs principaux: | , |
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| Format: | Journal article |
| Langue: | English |
| Publié: |
2007
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| _version_ | 1826280557217054720 |
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| author | Nikolov, N Pyber, L |
| author_facet | Nikolov, N Pyber, L |
| author_sort | Nikolov, N |
| collection | OXFORD |
| description | We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k>1, then G has a proper subgroup of index at most ck^2 for some absolute constant c, hence a product-free subset of size at least $|G| / c'k$. This answers a question of Gowers. |
| first_indexed | 2024-03-07T00:15:28Z |
| format | Journal article |
| id | oxford-uuid:7ab0d965-e610-46a7-ac0c-6e8b0cd2c6bb |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:15:28Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:7ab0d965-e610-46a7-ac0c-6e8b0cd2c6bb2022-03-26T20:45:40ZProduct decompositions of quasirandom groups and a Jordan type theoremJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:7ab0d965-e610-46a7-ac0c-6e8b0cd2c6bbEnglishSymplectic Elements at Oxford2007Nikolov, NPyber, LWe first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k>1, then G has a proper subgroup of index at most ck^2 for some absolute constant c, hence a product-free subset of size at least $|G| / c'k$. This answers a question of Gowers. |
| spellingShingle | Nikolov, N Pyber, L Product decompositions of quasirandom groups and a Jordan type theorem |
| title | Product decompositions of quasirandom groups and a Jordan type theorem |
| title_full | Product decompositions of quasirandom groups and a Jordan type theorem |
| title_fullStr | Product decompositions of quasirandom groups and a Jordan type theorem |
| title_full_unstemmed | Product decompositions of quasirandom groups and a Jordan type theorem |
| title_short | Product decompositions of quasirandom groups and a Jordan type theorem |
| title_sort | product decompositions of quasirandom groups and a jordan type theorem |
| work_keys_str_mv | AT nikolovn productdecompositionsofquasirandomgroupsandajordantypetheorem AT pyberl productdecompositionsofquasirandomgroupsandajordantypetheorem |