Page 87 - 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. 87
mož Moravec: Some Topics in the Theory of Finite Groups 75
The Correspondence Theorem says that if N is a normal subgroup of G then there is
a bijection between subgroups of G /N and subgroups of G containing N . The bijection
is canonical in the sense that all subgroups of G /N are of the form H /N , where H is a
subgroup of G containing N . This result enables construction of a composition series of
a finite group G as follows. Start with the series {1} G . If G is simple, we are done. Other-
wise there is a proper non-trivial normal subgroup N of G . Now we repeat the procedure
with {1} N and N G . More precisely, if we have Gi Gi +1 and the corresponding quo-
tient is not simple, then we choose (by the Correspondence Theorem) N /Gi Gi +1/Gi
with N = Gi and N = Gi +1. In this way we refine the series, and since the group is finite,
the procedure eventually results in a composition series of G . Given a composition series
of G as above, we have r simple groups Gi +1/Gi .
Theorem 3.2.32 (Jordan-Hölder Theorem) Any two composition series of a finite group
G give rise, up to order and isomorphism type, to the same list of composition factors.
PROOF. The proof is by induction on |G |. Let
G = G0 G1 G2 · · · Gr = {1}
and
G = H0 H1 G2 · · · Hs = {1}
be two composition series of G . If G1 = H1, then the parts of the series below this term
are two composition series of G1 and by induction they have the same length and com-
position factors. So assume from here on that G1 = H1. Let K2 = G1 ∩ H1. Let
K2 K3 · · · Kt = {1}
be a composition series of K2. The group G1H1 is a normal subgroup of G and G1 < G . It
follows that G = G1H1. Therefore G /G1 = G1H1/G1 =∼ H1/K2, and similarly also G /H1 ∼=
G1/K2. Thus
G1 G2 · · · Gr = {1}
and
G1 K2 K3 · · · Kt = {1}
are two composition series of G1 and hence they have the same length and same compo-
sition factors. A similar statement holds true for H1, so each of the given series for G has
the composition factors of K2 together with G /G1 and G /H1. Therefore the result holds.
Let us calculate a composition series of D32:
The Correspondence Theorem says that if N is a normal subgroup of G then there is
a bijection between subgroups of G /N and subgroups of G containing N . The bijection
is canonical in the sense that all subgroups of G /N are of the form H /N , where H is a
subgroup of G containing N . This result enables construction of a composition series of
a finite group G as follows. Start with the series {1} G . If G is simple, we are done. Other-
wise there is a proper non-trivial normal subgroup N of G . Now we repeat the procedure
with {1} N and N G . More precisely, if we have Gi Gi +1 and the corresponding quo-
tient is not simple, then we choose (by the Correspondence Theorem) N /Gi Gi +1/Gi
with N = Gi and N = Gi +1. In this way we refine the series, and since the group is finite,
the procedure eventually results in a composition series of G . Given a composition series
of G as above, we have r simple groups Gi +1/Gi .
Theorem 3.2.32 (Jordan-Hölder Theorem) Any two composition series of a finite group
G give rise, up to order and isomorphism type, to the same list of composition factors.
PROOF. The proof is by induction on |G |. Let
G = G0 G1 G2 · · · Gr = {1}
and
G = H0 H1 G2 · · · Hs = {1}
be two composition series of G . If G1 = H1, then the parts of the series below this term
are two composition series of G1 and by induction they have the same length and com-
position factors. So assume from here on that G1 = H1. Let K2 = G1 ∩ H1. Let
K2 K3 · · · Kt = {1}
be a composition series of K2. The group G1H1 is a normal subgroup of G and G1 < G . It
follows that G = G1H1. Therefore G /G1 = G1H1/G1 =∼ H1/K2, and similarly also G /H1 ∼=
G1/K2. Thus
G1 G2 · · · Gr = {1}
and
G1 K2 K3 · · · Kt = {1}
are two composition series of G1 and hence they have the same length and same compo-
sition factors. A similar statement holds true for H1, so each of the given series for G has
the composition factors of K2 together with G /G1 and G /H1. Therefore the result holds.
Let us calculate a composition series of D32:
