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Schreier index formula

(Theorem)

Let be a free group of finite rank, and let be a subgroup of finite index in . By the Nielsen-Schreier theorem, is free. The Schreier index formula states that

$\displaystyle \operatorname{rank}(H)=\vert F:H\vert\cdot(\operatorname{rank}(F)-1)+1.$

This implies more generally that if is a group generated by elements, then any subgroup of index in can be generated by at most elements.

"Schreier index formula" is owned by yark. [ full author list (2) | owner history (1) ]

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Cross-references: generated by, group generated by, implies, Nielsen-Schreier theorem, finite, finite rank, free group
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This is version 11 of Schreier index formula, born on 2003-09-05, modified 2004-09-15.
Object id is 4699, canonical name is ScheierIndexFormula.
Accessed 2280 times total.

Classification:

20E05 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Free nonabelian groups)

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