en>TakuyaMurata: /* See also */

2013-11-19T05:04:54Z

See also

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	Ed is ~~what people call me~~ and ~~my spouse doesn~~'~~t like it at all~~. ~~To climb~~ is ~~something I really appreciate performing~~. ~~Since he was 18 he~~'~~s been working as~~ an ~~information officer but he plans on changing it~~. ~~Ohio~~ is ~~where my home~~ is ~~but my husband desires us to move~~.<br><br>~~My web~~-~~site~~ .~~.. psychic chat online~~, ~~[http~~://~~cspl~~.~~postech~~.ac.kr/~~zboard~~/~~Membersonly~~/~~144571 http:~~//~~cspl~~.~~postech~~.ac.kr/~~zboard~~/~~Membersonly~~/~~144571~~],		In [[group theory]], a branch of [[mathematics]], '''Frattini's argument''' is an important [[Lemma (mathematics)\|lemma]] in the structure theory of [[finite group]]s. It is named after [[Giovanni Frattini]], who first used it in a paper from 1885 when defining the [[Frattini subgroup]] of a group.

			==Statement and proof==
			<blockquote>'''Frattini's Argument.''' If ''G'' is a finite group with normal subgroup ''H'', and if ''P'' is a [[Sylow p-subgroup\|Sylow ''p''-subgroup]] of ''H'', then

			:''G'' = ''N''<sub>''G''</sub>(''P'')''H'',

			where ''N''<sub>''G''</sub>(''P'') denotes the [[centralizer and normalizer\|normalizer]] of ''P'' in ''G''.</blockquote>

			'''Proof''': ''P'' is a Sylow ''p''-subgroup of ''H'', so every Sylow ''p''-subgroup of ''H'' is an ''H''-conjugate ''h''<sup>−1</sup>''Ph'' for some ''h'' ∈ ''H'' (see [[Sylow theorems]]). Let ''g'' be any element of ''G''. Since ''H'' is normal in ''G'', the subgroup ''g''<sup>−1</sup>''Pg'' is contained in ''H''. This means that ''g''<sup>−1</sup>''Pg'' is a Sylow ''p''-subgroup of ''H''. Then by the above, it must be ''H''-conjugate to ''P'': that is, for some ''h'' ∈ ''H''

			:''g''<sup>−1</sup>''Pg'' = ''h''<sup>−1</sup>''Ph'',

			so

			:''hg''<sup>−1</sup>''Pgh''<sup>−1</sup> = ''P'';

			thus

			:''gh''<sup>−1</sup> ∈ ''N''<sub>''G''</sub>(''P''),

			and therefore ''g'' ∈ ''N''<sub>''G''</sub>(''P'')''H''. But ''g'' ∈ ''G'' was arbitrary, so ''G'' = ''HN''<sub>''G''</sub>(''P'') = ''N''<sub>''G''</sub>(''P'')''H''. <math>\square</math>

			==Applications==
			* Frattini's argument can be used as part of a proof that any finite [[nilpotent group]] is a [[direct product of groups\|direct product]] of its Sylow subgroups.
			* By applying Frattini's argument to ''N''<sub>''G''</sub>(''N''<sub>''G''</sub>(''P'')), it can be shown that ''N''<sub>''G''</sub>(''N''<sub>''G''</sub>(''P'')) = ''N''<sub>''G''</sub>(''P'') whenever ''G'' is a finite group and ''P'' is a Sylow ''p''-subgroup of ''G''.
			* More generally, if a subgroup ''M'' ≤ ''G'' contains ''N''<sub>''G''</sub>(''P'') for some Sylow ''p''-subgroup ''P'' of ''G'', then ''M'' is self-normalizing, ''i.e.'' ''M'' = ''N''<sub>''G''</sub>(''M'').
			::Proof: ''M'' is normal in ''H'' := ''N''<sub>''G''</sub>(''M''), and ''P'' is a Sylow ''p''-subgroup of ''M'', so the Frattini argument applied to the group ''H'' with normal subgroup ''M'' and Sylow ''p''-subgroup ''P'' gives ''N''<sub>''H''</sub>(''P'')''M'' = ''H''. Since ''N''<sub>''H''</sub>(''P'') ≤ ''N''<sub>''G''</sub>(''P'') ≤ ''M'', one has the chain of inclusions ''M'' ≤ ''H'' = ''N''<sub>''H''</sub>(''P'')''M'' ≤ ''M'' ''M'' = ''M'', so ''M'' = ''H''. <math>\square</math>

			==References==


			* {{Cite book
			\| last = Hall
			\| first = Marshall
			\| authorlink =
			\| coauthors =
			\| title = The theory of groups
			\| publisher = Macmillan
			\| date = 1959
			\| location = New York, N.Y.
			\| pages =
			\| url =
			\| doi =
			\| id =
			\| isbn = }} (See Chapter 10, especially Section 10.4.)


			{{DEFAULTSORT:Frattini's Argument}}
			[[Category:Group theory]]
			[[Category:Lemmas]]
			[[Category:Articles containing proofs]]

189.106.53.131: /* n-connected space */

2012-01-02T22:28:38Z

n-connected space

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Ed is what people call me and my spouse doesn't like it at all. To climb is something I really appreciate performing. Since he was 18 he's been working as an information officer but he plans on changing it. Ohio is where my home is but my husband desires us to move.<br><br>My web-site ... psychic chat online, [http://cspl.postech.ac.kr/zboard/Membersonly/144571 http://cspl.postech.ac.kr/zboard/Membersonly/144571],

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en>TakuyaMurata: /* See also */

189.106.53.131: /* n-connected space */