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A subgroup H of finite index in a group G (finite or infinite) always contains a normal subgroup N (of G), also of finite index. In fact, if H has index n , then the index of N will be some divisor of n ! and a multiple of n ; indeed, N can be taken to be the kernel of the natural homomorphism from G to the permutation group of the left (or ...
The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the subgroup has finite index: if G is a free group of rank n (free on n generators), and H is a subgroup of finite index [G : H] = e, then H is free of rank + ().
A subgroup of finite index in a finitely generated group is always finitely generated, and the Schreier index formula gives a bound on the number of generators required. [2] In 1954, Albert G. Howson showed that the intersection of two finitely generated subgroups of a free group is again finitely generated.
More generally, a subgroup, , of finite index, , in contains a subgroup, , normal in and of index dividing ! called the normal core. In particular, if p {\displaystyle p} is the smallest prime dividing the order of G {\displaystyle G} , then every subgroup of index p {\displaystyle p} is normal.
Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if p is the lowest prime dividing the order of a finite group G , then any subgroup of index p (if such exists) is normal.
Every maximal subgroup of G has finite index in G. [19] The group G is finitely generated but not finitely presentable. [2] [20] The stabilizer of the level one vertices in in G (the subgroup of elements that act as identity on the strings 0 and 1), is generated by the following elements:
A subgroup of a profinite group is open if and only if it is closed and has finite index. According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely generated profinite group (that is, a profinite group that has a dense finitely generated subgroup) the subgroups of finite index are
In mathematics, the classification of finite simple groups (popularly called the enormous theorem [1] [2]) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits group is sometimes regarded as a sporadic group ...