GROUP OF ORDER 8 #4

GROUP #4

Dihedral(8)

The MAGMA library number for G is 3

GrpPC : G of order 8 = 2^3 PC-Relations: G.2^G.1 = G.2 * G.3

The center of G is abelian of type [ 2 ] .
The orders of the terms of the lower central series are [ 8, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 2, 8 ] .
The order of the Frattini subgroup is 2.
The exponent of G is 4.
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 4, 4 ] .
The orders of the centralizers of the maximal elementary abelian subgroups are [ 4, 4 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 8, 8 ] .

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x ] in degrees [ 1, 1, 2 ] , by the ideal generated by the relations
zy .

The Hilbert series for the cohomology ring is
1 / t2 -2t+ 1.
Its denominator factors as ( t-1 )2 .

The Krull dimension of the cohomology ring is 2.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
x , z2 + y2 .

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.3, G.2 * G.3 ]

The cohomology ring of E is a polynomial ring in the variables z , y .

The images of the generators of the cohomology of G restricted to E are
0 ,
z ,
zy + y2
in the cohomology of E.

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
z .

Subgroup E #2
Generated by [ G.1, G.3 ]

The cohomology ring of E is a polynomial ring in the variables z , y .

The images of the generators of the cohomology of G restricted to E are
z ,
0 ,
zy + y2
in the cohomology of E.

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
y .

The nilradical of the cohomology of G is zero.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

The group H is abelian of type [ 2, 2 ]

The cohomology ring of H is a polynomial ring with variables z , y in degrees [ 1, 1 ] .

The images of the generators of the cohomology of G restricted to H are
0 ,
z ,
zy + y2
in the cohomology of H.

The kernel of the restriction to H of the cohomology of G is generated by
z .

Maximal Subgroup H #2

The group H is abelian of type [ 2, 2 ]

The cohomology ring of H is a polynomial ring with variables z , y in degrees [ 1, 1 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
0 ,
zy + y2
in the cohomology of H.

The kernel of the restriction to H of the cohomology of G is generated by
y .

Maximal Subgroup H #3

The group H is abelian of type [ 4 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y ] , in degrees [ 1, 2 ] , by the ideal of relations
z2 .

The images of the generators of the cohomology of G restricted to H are
z ,
z ,
y
in the cohomology of H.

The kernel of the restriction to H of the cohomology of G is generated by
z + y .

The essential cohomology of G is zero.

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1

The kernel of the quotient is generated by G.3 .

The group Q is abelian of type [ 2, 2 ] .

The cohomology ring of Q is a polynomial ring with variables z , y in degrees [ 1, 1 ]

The images of the generators of the cohomology of Q inflated to G are
y ,
z
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy .

Action of Automorphisms

The groups of outer automorphisms of G has order 2, and is generated by 1 elements.

Automorphism #1

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

The images of the generators of the cohomology of G under the map induced by the automorphism are
y ,
z ,
x .

CHECKS

paramflag = true
qregflagflag = true
cmflag = true
essflag = true
centflag = true

THE COMPUTATION IS COMPLETE