Page 95 - 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 83
Conversely, suppose that G acts imprimitively on H \G . Let B be a block containing
the coset H , and denote K = {g ∈ G | B g = B }. Then H < K < G .
Proposition 3.3.5 Let G act primitively on X , and let N be a normal subgroup of G . Then
either N acts trivially on X , or N acts transitively on X .
PROOF. For x , y ∈ X put x ≡ y iff x h = y for some h ∈ N . For any g ∈ G we have
(x g )(g −1h g ) = y g . By normality, g −1h g ∈ N . Therefore x g ≡ y g , so ≡ is a G -congruence.
By primitivity, either all orbits have size 1 (i.e., N is in the kernel of the action), or there is
a single orbit (i.e., N acts transitively on X ).
Minimal and maximal subgroups
The above discussion on actions provides some useful descriptions of minimal and max-
imal subgroups of finite groups.
Lemma 3.3.6 A minimal normal subgroup of a finite group is isomorphic to the direct
product of a number of copies of a simple group.
PROOF. Let H be a minimal normal subgroup of G . By Lemma 3.2.4, H has no proper non-
tivial characteristic subgroups. Choose a minimal normal subgroup N of H of smallest
possible order. Consider all subgroups of H of the form N1 × · · · × Nn , where Ni H ,
Ni =∼ N . Let M be such group of largest possible order. If we show that M = H , then it
follows from here that N is simple. For, if K is a normal subgroup of N , then it is a normal
subgroup of M = N1 × · · · × Nn = G , and this contradicts the choice of N .
Thus it suffices to show that M is characteristic in H. Take φ ∈ Aut H . Then N φ ∼= N.
i
A straightforward argument shows that Niφ φ φ ≤ Niφ and
H. If N i ≤ M, then N i ∩ M
|Niφ ∩ M | < |N |. But Niφ ∩ M H , so the minimality of |N | shows Niφ ∩ M = {1}. The sub-
φ Niφ
group 〈M , N i 〉 = M × is of the same type like M but of larger order, a contradiction.
Thus M is characteristic in H .
Corollary 3.3.7 Let G be a finite solvable group. Then any maximal subgroup of G has
prime power index.
PROOF. Let H be a maximal subgroup of G and consider the action of G on H \G . By
Proposition 3.3.4, this action is primitive. The image of this action is a quotient of G ,
hence it is a solvable group. Therefore we may assume wlog that the action is faithful.
Let N be a minimal normal subgroup of G . Then N is an elementary abelian p -group by
Lemma 3.3.6. Snce G acts primitively, N acts transitively by Proposition 3.3.5. Using the
Conversely, suppose that G acts imprimitively on H \G . Let B be a block containing
the coset H , and denote K = {g ∈ G | B g = B }. Then H < K < G .
Proposition 3.3.5 Let G act primitively on X , and let N be a normal subgroup of G . Then
either N acts trivially on X , or N acts transitively on X .
PROOF. For x , y ∈ X put x ≡ y iff x h = y for some h ∈ N . For any g ∈ G we have
(x g )(g −1h g ) = y g . By normality, g −1h g ∈ N . Therefore x g ≡ y g , so ≡ is a G -congruence.
By primitivity, either all orbits have size 1 (i.e., N is in the kernel of the action), or there is
a single orbit (i.e., N acts transitively on X ).
Minimal and maximal subgroups
The above discussion on actions provides some useful descriptions of minimal and max-
imal subgroups of finite groups.
Lemma 3.3.6 A minimal normal subgroup of a finite group is isomorphic to the direct
product of a number of copies of a simple group.
PROOF. Let H be a minimal normal subgroup of G . By Lemma 3.2.4, H has no proper non-
tivial characteristic subgroups. Choose a minimal normal subgroup N of H of smallest
possible order. Consider all subgroups of H of the form N1 × · · · × Nn , where Ni H ,
Ni =∼ N . Let M be such group of largest possible order. If we show that M = H , then it
follows from here that N is simple. For, if K is a normal subgroup of N , then it is a normal
subgroup of M = N1 × · · · × Nn = G , and this contradicts the choice of N .
Thus it suffices to show that M is characteristic in H. Take φ ∈ Aut H . Then N φ ∼= N.
i
A straightforward argument shows that Niφ φ φ ≤ Niφ and
H. If N i ≤ M, then N i ∩ M
|Niφ ∩ M | < |N |. But Niφ ∩ M H , so the minimality of |N | shows Niφ ∩ M = {1}. The sub-
φ Niφ
group 〈M , N i 〉 = M × is of the same type like M but of larger order, a contradiction.
Thus M is characteristic in H .
Corollary 3.3.7 Let G be a finite solvable group. Then any maximal subgroup of G has
prime power index.
PROOF. Let H be a maximal subgroup of G and consider the action of G on H \G . By
Proposition 3.3.4, this action is primitive. The image of this action is a quotient of G ,
hence it is a solvable group. Therefore we may assume wlog that the action is faithful.
Let N be a minimal normal subgroup of G . Then N is an elementary abelian p -group by
Lemma 3.3.6. Snce G acts primitively, N acts transitively by Proposition 3.3.5. Using the
