Page 121 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 121
mož Moravec: Some Topics in the Theory of Finite Groups 109
5. G is the direct product of its Sylow subgroups.
PROOF. (1) ⇒ (2). Let G be nilpotent with class c . If H G , then HZi G HZi +1G since
Zi +1G /Zi G = Z (G /Zi G ). So, HZi G is the series proving subnormality of H .
(2) ⇒ (3). Let H = H0 H1 · · · Hn = G be the series proving subnormality of the
proper subgroup H . Let i be the smallest integer s.t. H = Hi . Then, H = Hi −1 Hi
NG (H ).
(3) ⇒ (4). If M < G is maximal, then M < NG (M ) implying NG (M ) = G .
(4) ⇒ (5). Assume P is a non-normal Sylow subgroup. Then NG (P) is proper and
therefore contained in a maximal subgroup M . Then M G contradicting Lemma 3.2.29.
Thus, Sylow p -subgroup is normal and consequently unique for each p . Their product
is clearly direct and equal to G .
(5) ⇒ (1). This follows since every p -group is nilpotent and direct sum of nilpotent
groups is nilpotent.
In the case of infinite groups, properties (2) to (5) are weaker than (1). Using the
above result, one can refine Corollary 3.3.7 as follows:
Corollary 3.5.15 A maximal subgroup M of a finite nilpotent group G has prime index.
PROOF. We known that M G , and |G : M | = p k by Corollary 3.3.7. If k > 1, then there
exists H < G containing M such that |H : M | = p which is a contradiction.
The Fitting Subgroup
Theorem 3.5.16 (Fitting) Let M and N be normal nilpotent subgroups of a group G . If c
and d are nilpotency classes of M and N , then L = M N is nilpotent of class ≤ c + d .
PROOF. By induction on i we show that
γi (L) = [X1, . . . , Xi ].
X j ∈{M ,N }
Taking i = c + d + 1 and noting that [A,G ] ≤ A for all A G , we conclude that each
[X1, . . . , Xi ] is contained in either γc+1(M ) or γd +1(N ), both of which equal to 1.
The subgroup Fit(G ) generated by all the normal nilpotent subgroups of a group G
is called the Fitting subgroup of G . If the group G is finite, then Fit(G ) is nilpotent. In
these cases, Fit(G ) is the unique largest normal nilpotent subgroup of G . Note also that
Fit(G ) = 1 if and only if G is semisimple.
Let N ≤ H ≤ G and N G . Define CG (H /N ) = {g ∈ G : [H , g ] ≤ N }. Clearly CG (H /N ) ≤
G.
5. G is the direct product of its Sylow subgroups.
PROOF. (1) ⇒ (2). Let G be nilpotent with class c . If H G , then HZi G HZi +1G since
Zi +1G /Zi G = Z (G /Zi G ). So, HZi G is the series proving subnormality of H .
(2) ⇒ (3). Let H = H0 H1 · · · Hn = G be the series proving subnormality of the
proper subgroup H . Let i be the smallest integer s.t. H = Hi . Then, H = Hi −1 Hi
NG (H ).
(3) ⇒ (4). If M < G is maximal, then M < NG (M ) implying NG (M ) = G .
(4) ⇒ (5). Assume P is a non-normal Sylow subgroup. Then NG (P) is proper and
therefore contained in a maximal subgroup M . Then M G contradicting Lemma 3.2.29.
Thus, Sylow p -subgroup is normal and consequently unique for each p . Their product
is clearly direct and equal to G .
(5) ⇒ (1). This follows since every p -group is nilpotent and direct sum of nilpotent
groups is nilpotent.
In the case of infinite groups, properties (2) to (5) are weaker than (1). Using the
above result, one can refine Corollary 3.3.7 as follows:
Corollary 3.5.15 A maximal subgroup M of a finite nilpotent group G has prime index.
PROOF. We known that M G , and |G : M | = p k by Corollary 3.3.7. If k > 1, then there
exists H < G containing M such that |H : M | = p which is a contradiction.
The Fitting Subgroup
Theorem 3.5.16 (Fitting) Let M and N be normal nilpotent subgroups of a group G . If c
and d are nilpotency classes of M and N , then L = M N is nilpotent of class ≤ c + d .
PROOF. By induction on i we show that
γi (L) = [X1, . . . , Xi ].
X j ∈{M ,N }
Taking i = c + d + 1 and noting that [A,G ] ≤ A for all A G , we conclude that each
[X1, . . . , Xi ] is contained in either γc+1(M ) or γd +1(N ), both of which equal to 1.
The subgroup Fit(G ) generated by all the normal nilpotent subgroups of a group G
is called the Fitting subgroup of G . If the group G is finite, then Fit(G ) is nilpotent. In
these cases, Fit(G ) is the unique largest normal nilpotent subgroup of G . Note also that
Fit(G ) = 1 if and only if G is semisimple.
Let N ≤ H ≤ G and N G . Define CG (H /N ) = {g ∈ G : [H , g ] ≤ N }. Clearly CG (H /N ) ≤
G.
