3: Series
Given subgroups A, B of a group G we obtain the subgroup
[A,B] = {[a,b] | aeA.be B).
In particular, [G, G] is the derived group of G. We define the derived se-
ries of G to be the most rapidly descending series with abelian quotients
(factors), namely
G<0) = G, (Vi > 1) G<l) = [Gt*"1', G^" 1 )].
We say that G is soluble of derived length n if n is the least integer with
GW = {1}.
Similarly, the most rapidly descending central series and the most
rapidly ascending central series of G are the lower central series and the
upper central series, defined by
Zo = {l}, (V»>1) Zi/Zi_1=Z(G/Zi-1)
respectively. The lower central series reaches {1} in a finite number of
steps if and only if the upper central series reaches G in a finite number
of steps. In this case G is said to be nilpotent, and the number of factors
in either series is the class of G.
Every subgroup H of a nilpotent group G is subnormal, in the sense
that there is a series
H = Ho < Hi < • • • < Hr = G.
The final type of series with which we assume the reader is familiar
is called a composition series^ This is a subnormal series from {1} into
which no further terms can be properly inserted.
, Book 5 Groups
3.1 Let G be a group. Establish each of the following results concerning
commutators.
(a) If S < G and T < G then [5, T] = [T, S].
(b) If H < G and K < G then [H, K] < H n K. What does this imply
when HC\K = {1}1
(c) If x, y, z e G then
Deduce that if H, K, L are normal subgroups of G then
(d) Define [a,6,c,] = [[a,6],c]. Prove that
[a, 6c] = [a,c][a,6][a,6,c]
and that
\ab,c] = [a,c}[a,c,b}[b,c}.
3.2 Find the upper and lower central series of G = Qs x (?2 and show that
they do not coincide. Show, however, that the upper and lower central
series of Qs do coincide.
3.3 Prove that if G is generated by its subnormal abelian subgroups then any
quotient group of G is generated by its subnormal abelian subgroups.
Show that every subgroup of a nilpotent group is subnormal. Deduce
that a nilpotent group is generated by its subnormal abelian subgroups.
3.4 Let A, B, C be subgroups of a group G with B < A. Prove that
AnG ^ B{ApC)
Br\C'
If, in addition, C<G prove that
AC
BC "~ B{AnC)'
Use the above results to show that if H is a soluble group then every
subgroup and every quotient group of H is soluble.
Prove that if K is a group with H < K and both H and K/H are
soluble then K is soluble.
Let G be a group with normal subgroups A and B such that G/A and
G/B are soluble. Show that A/(A n B) is soluble and deduce that so
also is G/{AnB).
16
Given subgroups A, B of a group G we obtain the subgroup
[A,B] = {[a,b] | aeA.be B).
In particular, [G, G] is the derived group of G. We define the derived se-
ries of G to be the most rapidly descending series with abelian quotients
(factors), namely
G<0) = G, (Vi > 1) G<l) = [Gt*"1', G^" 1 )].
We say that G is soluble of derived length n if n is the least integer with
GW = {1}.
Similarly, the most rapidly descending central series and the most
rapidly ascending central series of G are the lower central series and the
upper central series, defined by
Zo = {l}, (V»>1) Zi/Zi_1=Z(G/Zi-1)
respectively. The lower central series reaches {1} in a finite number of
steps if and only if the upper central series reaches G in a finite number
of steps. In this case G is said to be nilpotent, and the number of factors
in either series is the class of G.
Every subgroup H of a nilpotent group G is subnormal, in the sense
that there is a series
H = Ho < Hi < • • • < Hr = G.
The final type of series with which we assume the reader is familiar
is called a composition series^ This is a subnormal series from {1} into
which no further terms can be properly inserted.
, Book 5 Groups
3.1 Let G be a group. Establish each of the following results concerning
commutators.
(a) If S < G and T < G then [5, T] = [T, S].
(b) If H < G and K < G then [H, K] < H n K. What does this imply
when HC\K = {1}1
(c) If x, y, z e G then
Deduce that if H, K, L are normal subgroups of G then
(d) Define [a,6,c,] = [[a,6],c]. Prove that
[a, 6c] = [a,c][a,6][a,6,c]
and that
\ab,c] = [a,c}[a,c,b}[b,c}.
3.2 Find the upper and lower central series of G = Qs x (?2 and show that
they do not coincide. Show, however, that the upper and lower central
series of Qs do coincide.
3.3 Prove that if G is generated by its subnormal abelian subgroups then any
quotient group of G is generated by its subnormal abelian subgroups.
Show that every subgroup of a nilpotent group is subnormal. Deduce
that a nilpotent group is generated by its subnormal abelian subgroups.
3.4 Let A, B, C be subgroups of a group G with B < A. Prove that
AnG ^ B{ApC)
Br\C'
If, in addition, C<G prove that
AC
BC "~ B{AnC)'
Use the above results to show that if H is a soluble group then every
subgroup and every quotient group of H is soluble.
Prove that if K is a group with H < K and both H and K/H are
soluble then K is soluble.
Let G be a group with normal subgroups A and B such that G/A and
G/B are soluble. Show that A/(A n B) is soluble and deduce that so
also is G/{AnB).
16
