Easy question on L-theory

Question

I'm reading the introductory book on L-theory, Algebraic and geometric surgery by Andrew Ranicki, available at 1.

Here is the statement in the page 254 Example 11.18.

If $G$ is a group without $2$-torsion, then (there is) a decomposition $ G=\lbrace 1\rbrace\cup S\cup S^{-1}$ which determines a $\mathbb{Z}[\mathbb{Z}/2]$-module splitting of $\mathbb{Z}[G]=\mathbb{Z}\oplus(\mathbb{Z}[S]\oplus\mathbb{Z}[S^{-1}])$.

Here, how can I decompose $G$ into $G=\lbrace 1\rbrace\cup S\cup S^{-1}$ for $G$ a group without $2$-torsion?

If $G$ has no $2$-torsion, then inversion has only one fixed point, so... — Qiaochu Yuan, Mar 29 '11 at 16:11
Qiaochu, I know there is only one order 2 element, but is it true that there is the natural way to split $G-\lbrace 1\rbrace$ into two pieces for such arbitrary group $G$? — user8484, Mar 29 '11 at 17:49

score 2 · Accepted Answer · answered Mar 29 '11 at 18:57

2

Since there's no $g\in G$ such that $g^2=1$, the set $\{g\in G\,|\,g\neq1\}$ is disjoint union of sets of the form $\{g,g^{-1}\}$ consisting of exactly two elements.

Let $S$ be a subset of $G$ containing exactly one element for each such pair (guess you need the axiom of choice here). The stated decomposition follows at once.

answered Mar 29 '11 at 18:57

Andrea Mori

26,969

This answer is what I want. Thanks. – user8484 Mar 29 '11 at 19:00

Easy question on L-theory

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