2: Automorphisms and Sylow theory
An isomorphism / : G —> G is called an automorphism on G. The auto-
morphisms on a group G form, under composition of mappings, a group
AutG. Conjugation by a fixed element g of G, namely the mapping
<pg : G —> G described by x F-> ^ ( E ) = g~1zg, is an automorphism on
G. The inner automorphism group Inn G = {cpg | g G G} is a normal
subgroup of Aut G, and the quotient group Aut G/Inn G is called the
outer automorphism group of G. For example, the cyclic group Cn (be-
ing abelian) has trivial inner automorphism group, and & : Cn —> Cn
given by &(g) = g~l is an (outer) automorphism of order 2. A subgroup
/ / of a group G is normal if and only if $(H) C H for every # G Inn G,
and is called characteristic if $(H) C iiT for every # G Aut G.
For finite groups, the converse of Lagrange's theorem is false. How-
ever, a partial converse is provided by the important theorems of Sylow.
A group P is called a p-group if every element has order a power of p
for a fixed prime p. In this case, if P is finite, \P\ is also a power of p.
If G is a group with \G\ = pnk where k is coprime to p then a subgroup
of order pn is called a Sylow p-subgroup. In this situation we have the
following results, with which we assume the reader is familiar :
(a) G has a subgroup of order pm for every m < n;
(b) every p-subgroup of G is contained in a Sylow p-subgroup;
(c) any two Sylow ^-subgroups are conjugate in G;
(d) the number of Sylow p-subgroups of G is congruent to 1 modulo p
and divides |G|.
2.1 Let p be a prime. Use the class equation to show that every finite p-
group has a non-trivial centre. Deduce that all groups of order p2 are
abelian.
, 2: Automorphisms and Sylow theory
List all the groups of order 9.
2.2 Let G be a group and let $ 6 Aut G. If A and B are subgroups of G
prove that $(AnB) is a subgroup of flAntiB. Is it necessarily true that
2.3 Let G be a group and let Inn G be the group of inner automorphisms on
G. Prove that InnCr is a normal subgroup of Aut G and. that
InnG~G/Z{G).
The two non-abelian groups of order 8 are the dihedral group Ds with
presentation
D8 = (a,b 1 a2 = 1, b4 = 1, a~Ha = 6" 1 )
and the quaternion group Q8 with presentation
Qs = {xiy | x4 = l, x2^y2, y-'xy^x'1).
Show that Z{D8) = (b2) ~C2 and Z(Q8) = {x2) ~ C 2 . Deduce that
Inn Dg ~ Inn Qg •
2.4 Let <3 be a group with the property that it cannot be decomposed into
the direct product of two non-trivial subgroups. Does every subgroup
of G have this property? Does every quotient group of G have this
property?
2.5 If G is a group such that G/Z(G) is cyclic, prove that G is abelian.
Deduce that a group with a cyclic automorphism group is necessarily
abelian.
2.6 Find the automorphism group of the symmetric group 63.
2.7 Prove that C2 x C2 and £3 have isomorphic automorphism groups.
2.8 Let G be a group with Z[G) = {1}. Prove that Z(Aut G) = {1}. Is the
converse true in general?
2.9 Let Z p denote the field of integers modulo p where p is a prime, and let
Zp be an ra-dimensional vector space over Z p . Prove that the additive
group of Zp is isomorphic to the group
Gpx Cpx • • • x Cp
consisting of n copies of Cp.
Show that every element of Aut G corresponds to an invertible linear
transformation on Z£.
Deduce that
2.10 Find all groups G with Aut G = {1}.
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An isomorphism / : G —> G is called an automorphism on G. The auto-
morphisms on a group G form, under composition of mappings, a group
AutG. Conjugation by a fixed element g of G, namely the mapping
<pg : G —> G described by x F-> ^ ( E ) = g~1zg, is an automorphism on
G. The inner automorphism group Inn G = {cpg | g G G} is a normal
subgroup of Aut G, and the quotient group Aut G/Inn G is called the
outer automorphism group of G. For example, the cyclic group Cn (be-
ing abelian) has trivial inner automorphism group, and & : Cn —> Cn
given by &(g) = g~l is an (outer) automorphism of order 2. A subgroup
/ / of a group G is normal if and only if $(H) C H for every # G Inn G,
and is called characteristic if $(H) C iiT for every # G Aut G.
For finite groups, the converse of Lagrange's theorem is false. How-
ever, a partial converse is provided by the important theorems of Sylow.
A group P is called a p-group if every element has order a power of p
for a fixed prime p. In this case, if P is finite, \P\ is also a power of p.
If G is a group with \G\ = pnk where k is coprime to p then a subgroup
of order pn is called a Sylow p-subgroup. In this situation we have the
following results, with which we assume the reader is familiar :
(a) G has a subgroup of order pm for every m < n;
(b) every p-subgroup of G is contained in a Sylow p-subgroup;
(c) any two Sylow ^-subgroups are conjugate in G;
(d) the number of Sylow p-subgroups of G is congruent to 1 modulo p
and divides |G|.
2.1 Let p be a prime. Use the class equation to show that every finite p-
group has a non-trivial centre. Deduce that all groups of order p2 are
abelian.
, 2: Automorphisms and Sylow theory
List all the groups of order 9.
2.2 Let G be a group and let $ 6 Aut G. If A and B are subgroups of G
prove that $(AnB) is a subgroup of flAntiB. Is it necessarily true that
2.3 Let G be a group and let Inn G be the group of inner automorphisms on
G. Prove that InnCr is a normal subgroup of Aut G and. that
InnG~G/Z{G).
The two non-abelian groups of order 8 are the dihedral group Ds with
presentation
D8 = (a,b 1 a2 = 1, b4 = 1, a~Ha = 6" 1 )
and the quaternion group Q8 with presentation
Qs = {xiy | x4 = l, x2^y2, y-'xy^x'1).
Show that Z{D8) = (b2) ~C2 and Z(Q8) = {x2) ~ C 2 . Deduce that
Inn Dg ~ Inn Qg •
2.4 Let <3 be a group with the property that it cannot be decomposed into
the direct product of two non-trivial subgroups. Does every subgroup
of G have this property? Does every quotient group of G have this
property?
2.5 If G is a group such that G/Z(G) is cyclic, prove that G is abelian.
Deduce that a group with a cyclic automorphism group is necessarily
abelian.
2.6 Find the automorphism group of the symmetric group 63.
2.7 Prove that C2 x C2 and £3 have isomorphic automorphism groups.
2.8 Let G be a group with Z[G) = {1}. Prove that Z(Aut G) = {1}. Is the
converse true in general?
2.9 Let Z p denote the field of integers modulo p where p is a prime, and let
Zp be an ra-dimensional vector space over Z p . Prove that the additive
group of Zp is isomorphic to the group
Gpx Cpx • • • x Cp
consisting of n copies of Cp.
Show that every element of Aut G corresponds to an invertible linear
transformation on Z£.
Deduce that
2.10 Find all groups G with Aut G = {1}.
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