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 July 19th, 2014, 11:56 PM #1 Newbie   Joined: Jul 2014 From: Czech republic Posts: 1 Thanks: 0 Saturated subgroup Hi, let $\displaystyle G$ be an l-group and $\displaystyle g\in G$. $\displaystyle S(g)$ denotes the minimal saturated subgroup that contains $\displaystyle g$. Proposition: $\displaystyle S(g)$ is the subgroup of $\displaystyle G$ generated by the components of $\displaystyle g$. Proof: Let $\displaystyle A(g)$ be the subgroup of $\displaystyle G$ generated by the components of $\displaystyle g$. WLOG $\displaystyle 1\leq g$ ($\displaystyle 1$ denotes the neutral element of the group). And so on. My problem: I don't understand why I can write the WLOG part. While I know that $\displaystyle S(g)=S(|g|)$, I am not sure if $\displaystyle A(g)\supseteq A(|g|)$. For this to be true, it would have to be possible to express any component of $\displaystyle |g|$ in terms of components of $\displaystyle g$ and the group (not lattice) operations. The inclusion $\displaystyle A(g)\subseteq A(|g|)$ is clear to me. Thank you. Tags saturated, subgroup Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post tinynerdi Abstract Algebra 4 January 27th, 2010 09:02 PM HairOnABiscuit Abstract Algebra 3 November 28th, 2009 11:10 AM envision Abstract Algebra 3 October 4th, 2009 10:37 PM envision Abstract Algebra 1 October 4th, 2009 03:24 AM Mathworm Abstract Algebra 1 April 28th, 2007 06:11 AM

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