Page 119 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
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mož Moravec: Some Topics in the Theory of Finite Groups 107
Now, let G be nilpotent of class c , let d be the derived length and let 2i c + 1. Then,
G (i ) γ2i (G ) γc+1(G ) = 1. Since the smallest such i is log2 c + 1, it follows that
d log2 c + 1.
Here is a sample computation of lower and upper central series of a group:
gap> G := SmallGroup(128, 50);;
gap> NilpotencyClassOfGroup(G);
4
gap> DerivedLength(G);
2
gap> LowerCentralSeriesOfGroup(G);
[, Group([ f3, f5, f7 ]),
Group([ f5, f7 ]), Group([ f7 ]), Group([ of ... ]) ]
gap> UpperCentralSeriesOfGroup(G);
[ Group([ f6, f7, f5, f3, f4, f1, f2 ]), Group([ f6, f7, f5, f3, f4 ]),
Group([ f6, f7, f5 ]), Group([ f6, f7 ]), Group([ ]) ]
Unitriangular groups
Here is a ring-theoretic source of examples of nilpotent groups. Let S be a ring with iden-
tity and N a subring. Write N (i ) for the set of all sums of products of i elements of N for
i > 0, which is necessarily a subring. If N (i ) = 0 for some i > 0, then N is called nilpotent.
Assume N (n) = 0 and let U be the set of all elements of the form 1 + x for x ∈ N . Then U
is a group with respect to the ring multiplication, i.e.
(1 + x )(1 + y ) = 1 + (x + y + x y )
and
(1 + x )−1 = 1 + (−x + x 2 − · · · + (−x )n−1).
Define Ui = {1 + x | x ∈ N (i )} and observe that Ui is an increasing series of subgroups. We
want to show that this is actually a central series of U . Let x ∈ N (r ) and y ∈ N (s), then
[1 + x , 1 + y ] = (1 + x + y + y x )−1(1 + x + y + x y ).
We let u = x + y + x y and v = x + y + y x :
[1 + x , 1 + y ] = (1 − v + v 2 − · · · + (−v )n−1)(1 + u ) =
1 + (1 − v + v 2 − · · · + (−v )n−2)(u − v ) + (−v )n−1u .
Now, u −v = x y −y x ∈ N (r +s ) and (−v )n−1u = 0. We have thus shown that [Ur ,Us ] Ur +s
implying that U is nilpotent of class no more than n − 1.
For an even more concrete example, let us take S to be the ring of all n × n matrices
over a commutative ring with identity R. Further, let N be the subring of all strictly upper
Now, let G be nilpotent of class c , let d be the derived length and let 2i c + 1. Then,
G (i ) γ2i (G ) γc+1(G ) = 1. Since the smallest such i is log2 c + 1, it follows that
d log2 c + 1.
Here is a sample computation of lower and upper central series of a group:
gap> G := SmallGroup(128, 50);;
gap> NilpotencyClassOfGroup(G);
4
gap> DerivedLength(G);
2
gap> LowerCentralSeriesOfGroup(G);
[
Group([ f5, f7 ]), Group([ f7 ]), Group([
gap> UpperCentralSeriesOfGroup(G);
[ Group([ f6, f7, f5, f3, f4, f1, f2 ]), Group([ f6, f7, f5, f3, f4 ]),
Group([ f6, f7, f5 ]), Group([ f6, f7 ]), Group([ ]) ]
Unitriangular groups
Here is a ring-theoretic source of examples of nilpotent groups. Let S be a ring with iden-
tity and N a subring. Write N (i ) for the set of all sums of products of i elements of N for
i > 0, which is necessarily a subring. If N (i ) = 0 for some i > 0, then N is called nilpotent.
Assume N (n) = 0 and let U be the set of all elements of the form 1 + x for x ∈ N . Then U
is a group with respect to the ring multiplication, i.e.
(1 + x )(1 + y ) = 1 + (x + y + x y )
and
(1 + x )−1 = 1 + (−x + x 2 − · · · + (−x )n−1).
Define Ui = {1 + x | x ∈ N (i )} and observe that Ui is an increasing series of subgroups. We
want to show that this is actually a central series of U . Let x ∈ N (r ) and y ∈ N (s), then
[1 + x , 1 + y ] = (1 + x + y + y x )−1(1 + x + y + x y ).
We let u = x + y + x y and v = x + y + y x :
[1 + x , 1 + y ] = (1 − v + v 2 − · · · + (−v )n−1)(1 + u ) =
1 + (1 − v + v 2 − · · · + (−v )n−2)(u − v ) + (−v )n−1u .
Now, u −v = x y −y x ∈ N (r +s ) and (−v )n−1u = 0. We have thus shown that [Ur ,Us ] Ur +s
implying that U is nilpotent of class no more than n − 1.
For an even more concrete example, let us take S to be the ring of all n × n matrices
over a commutative ring with identity R. Further, let N be the subring of all strictly upper
