Editing P-group (section)

===Unitriangular matrix groups===

The Sylow subgroups of [[general linear group]]s are another fundamental family of examples. Let ''V'' be a vector space of dimension ''n'' with basis { ''e''<sub>1</sub>, ''e''<sub>2</sub>, ..., ''e''<sub>''n''</sub> } and define ''V''<sub>''i''</sub> to be the vector space generated by { ''e''<sub>''i''</sub>, ''e''<sub>''i''+1</sub>, ..., ''e''<sub>''n''</sub> } for 1 ≤ ''i'' ≤ ''n'', and define ''V''<sub>''i''</sub> = 0 when ''i'' &gt; ''n''. For each 1 ≤ ''m'' ≤ ''n'', the set of invertible linear transformations of ''V'' which take each ''V''<sub>''i''</sub> to ''V''<sub>''i''+''m''</sub> form a subgroup of Aut(''V'') denoted ''U''<sub>''m''</sub>. If ''V'' is a vector space over '''Z'''/''p'''''Z''', then ''U''<sub>1</sub> is a Sylow ''p''-subgroup of Aut(''V'') = GL(''n'', ''p''), and the terms of its [[lower central series]] are just the ''U''<sub>''m''</sub>. In terms of matrices, ''U''<sub>''m''</sub> are those upper triangular matrices with 1s one the diagonal and 0s on the first ''m''−1 superdiagonals. The group ''U''<sub>1</sub> has order ''p''<sup>''n''·(''n''−1)/2</sup>, nilpotency class ''n'', and exponent ''p''<sup>''k''</sup> where ''k'' is the least integer at least as large as the base ''p'' [[logarithm]] of ''n''.