78.146.15.4: /* Distance ratios */ clearer

2013-03-26T20:00:45Z

Distance ratios: clearer

← Older revision		Revision as of 22:00, 26 March 2013
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	~~Claude~~ is ~~her title~~ and ~~she totally digs~~ that ~~title~~. ~~Alabama~~ is ~~where he~~ and ~~his spouse live~~ and ~~he has everything that he requirements there~~. One of the ~~things I adore most~~ is ~~greeting card gathering but I don~~'~~t have~~ the ~~time recently~~. ~~Interviewing~~ is ~~what she does~~.<br><br>~~Also visit my web-site;~~ [~~http~~://~~ernrduo~~.~~woobi~~.Co.kr/xe/~~?document_srl~~=~~582849 ernrduo~~.~~woobi~~.Co.kr]		'''[[Goursat]]'s lemma''' is an [[algebra]]ic [[theorem]] about [[subgroup]]s of the [[Direct product of groups\|direct product]] of two [[Group (mathematics)\|groups]].

			It can be stated as follows.
			:Let <math>G</math>, <math>G'</math> be groups, and let <math>H</math> be a subgroup of <math>G\times G'</math> such that the two [[projection (mathematics)\|projections]] <math>p_1: H\rightarrow G</math> and <math>p_2: H\rightarrow G'</math> are [[surjective]] (i.e., <math>H</math> is a [[subdirect product]] of <math>G</math> and <math>G'</math>). Let <math>N</math> be the kernel of <math>p_2</math> and <math>N'</math> the [[Kernel (algebra)\|kernel]] of <math>p_1</math>. One can identify <math>N</math> as a [[normal subgroup]] of <math>G</math>, and <math>N'</math> as a normal subgroup of <math>G'</math>. Then the image of <math>H</math> in <math>G/N\times G'/N'</math> is the [[graph of a function\|graph]] of an [[isomorphism]] <math>G/N\approx G'/N'</math>.

			An immediate consequence of this is that the subdirect product of two groups can be described as a [[Direct product of groups#Fiber products\|fiber product]] and vice versa.

			== Proof of Goursat's lemma ==

			Before proceeding with the [[Mathematical proof\|proof]], <math>N</math> and <math>N'</math> are shown to be normal in <math>G \times \{e'\}</math> and <math>\{e\} \times G'</math>, respectively. It is in this sense that <math>N</math> and <math>N'</math> can be identified as normal in ''G'' and ''G''', respectively.

			Since <math>p_2</math> is a [[homomorphism]], its kernel ''N'' is normal in ''H''. Moreover, given <math>g \in G</math>, there exists <math>h=(g,g') \in H</math>, since <math>p_1</math> is surjective. Therefore, <math>p_1(N)</math> is normal in ''G'', viz:
			:<math>gp_1(N)=p_1(h)p_1(N)=p_1(hN)=p_1(Nh)=p_1(N)g</math>.
			It follows that <math>N</math> is normal in <math>G \times \{e'\}</math> since
			: <math>(g,e')N = (g,e')(p_1(N) \times \{e'\}) = gp_1(N) \times \{e'\} = p_1(N)g \times \{e'\} = (p_1(N) \times \{e'\})(g,e')=N(g,e')</math>.

			The proof that <math>N'</math> is normal in <math>\{e\} \times G'</math> proceeds in a similar manner.

			Given the identification of <math>G</math> with <math>G \times \{e'\}</math>, we can write <math>G/N</math> and <math>gN</math> instead of <math>(G \times \{e'\})/N</math> and <math>(g,e')N</math>, <math>g \in G</math>. Similarly, we can write <math>G'/N'</math> and <math>g'N'</math>, <math>g' \in G'</math>.

			On to the proof. Consider the map <math>H \rightarrow G/N \times G'/N'</math> defined by <math>(g,g') \mapsto (gN, g'N')</math>. The image of <math>H</math> under this map is <math>\{(gN,g'N') \| (g,g') \in H \}</math>. This [[Relation (mathematics)\|relation]] is the graph of a [[well-defined]] function <math>G/N \rightarrow G'/N'</math> provided <math>gN=N \Rightarrow g'N'=N'</math>, essentially an application of the [[vertical line test]].

			Since <math>gN=N</math> (more properly, <math>(g,e')N=N</math>), we have <math>(g,e') \in N \subset H</math>. Thus <math>(e,g') = (g,g')(g^{-1},e') \in H</math>, whence <math>(e,g') \in N'</math>, that is, <math>g'N'=N'</math>. Note that by symmetry, it is immediately clear that <math>g'N'=N' \Rightarrow gN=N</math>, i.e., this function also passes the [[horizontal line test]], and is therefore [[injective function\|one-to-one]]. The fact that this function is a surjective group homomorphism follows directly.

			== References ==

			* [[Ken Ribet\|Kenneth A. Ribet]] (Autumn 1976), "[[Galois]] [[Group action\|Action]] on Division Points of [[Abelian Variety\|Abelian Varieties]] with Real Multiplications", ''[[American Journal of Mathematics]]'', Vol. 98, No. 3, 751–804.

			[[Category:Algebra]]
			[[Category:Lemmas]]
			[[Category:Articles containing proofs]]

en>David Eppstein: /* External links */ not stub

2012-09-02T01:00:59Z

External links: not stub

New page

Claude is her title and she totally digs that title. Alabama is where he and his spouse live and he has everything that he requirements there. One of the things I adore most is greeting card gathering but I don't have the time recently. Interviewing is what she does.<br><br>Also visit my web-site; [http://ernrduo.woobi.Co.kr/xe/?document_srl=582849 ernrduo.woobi.Co.kr]

Isodynamic point - Revision history

78.146.15.4: /* Distance ratios */ clearer

en>David Eppstein: /* External links */ not stub