Do subgroups have "two sided bases"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:40:14Z http://mathoverflow.net/feeds/question/6647 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6647/do-subgroups-have-two-sided-bases Do subgroups have "two sided bases"? Dave Penneys 2009-11-24T03:34:32Z 2009-11-24T18:39:27Z <p>Let $H\leq G$ be an inclusion of finite groups. Define a map $E\colon \mathbb{C}[G]\to \mathbb{C}[H]$ to be the $\mathbb{C}$-linear extension of $$E(g)=\begin{cases} g &amp;\text{if } g\in H\\ 0 &amp;\text{else,} \end{cases}$$ i.e., $E$ is the projection onto $\mathbb{C}[H]$. A finite subset $B\subset \mathbb{C}[G]$ will be called a left basis for $G$ over $H$ if $$x=\sum\limits_{b\in B} b E(b^\ast x)$$ for all $x\in \mathbb{C}[G]$, where $\ast$ is the anti-linear extension of the map $g\mapsto g^{-1}$. For an example, take $B$ to be a set of left-coset representatives. Similarly, we can define a right basis to be a finite subset $B\subset \mathbb{C}[G]$ such that $$x=\sum\limits_{b\in B} E(x b^\ast)b$$ for all $x\in\mathbb{C}[G]$.</p> <p>Note that there exist groups for which there is a basis which is both a left and right basis, but $H$ is not a normal subgroup of $G$. One can take the subgroup of the symmetric group $S_n$ ($n\geq 3$) which fixes $1$. Then a set of left and right coset representatives is given by $$\{ (1 j)|j=1,\dots,n\}.$$ Does there always exist a basis which is both a left and right basis, or are there inclusions of groups for which there is no simultaneous left and right basis?</p> <p>The motivation for this question is another question from subfactor theory: if $N\subset M$ is a finite index, extremal $II_1$-subfactor, does there always exist a Pimsner-Popa basis which is both a left and right basis? The subgroup subfactor is an example of such a subfactor, and the question posed above is a watered-down version of the subfactor question, where perhaps an answer is already known or more easily obtainable.</p> http://mathoverflow.net/questions/6647/do-subgroups-have-two-sided-bases/6652#6652 Answer by Mariano Suárez-Alvarez for Do subgroups have "two sided bases"? Mariano Suárez-Alvarez 2009-11-24T04:21:16Z 2009-11-24T04:28:23Z <p>If $H$ is a subgroup of finite index in a group $G$, there is a subset $\mathcal B$ of $G$ which serves both as a set of representatives for the left cosets of $H$ in $G$ and as a set of representatives for the right cosets of $H$ in $G$. (See, for example, Theorem 3, §3, Chap. I, in the book <em>The Theory of groups</em> by H. Zassenhaus) That should do it, no?</p>