View source for Rank of a group

''(general and special rank of a group)''

A notion from group theory. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774401.png" /> has finite general rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774402.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774403.png" /> is the minimal number such that any finitely-generated subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774404.png" /> is contained in a subgroup having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774405.png" /> generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774406.png" />. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774407.png" /> has finite special rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774408.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r0774409.png" /> is the minimal number such that any finitely-generated subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r07744010.png" /> has a system of generators of at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077440/r07744011.png" /> elements. If no such finite number exists, then the general (special) rank of the group is said to be infinite.

The general rank of a group is smaller than or equal to its special rank. There are groups with finite general rank (even equal to two) and infinite special rank. Such is, for instance, the countable symmetric group. For an Abelian group the general and special rank coincide with its Prüfer rank (see [[Abelian group|Abelian group]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR></table>