GROUP OF ORDER 16 #9

GROUP #9

The MAGMA library number for G is 3

GrpPC : G of order 16 = 2^4 PC-Relations: G.1^2 = G.3 * G.4, G.2^2 = G.3, G.2^G.1 = G.2 * G.4

The center of G is abelian of type [ 2, 2 ] .
The orders of the terms of the lower central series are [ 16, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 4, 16 ] .
The order of the Frattini subgroup is 4.
The exponent of G is 4.
G has a unique maximal elementary abelian subgroup which is normal and has order 8. Its centralizer has order 8.

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w, v ] in degrees [ 1, 1, 2, 2, 2 ] , by the ideal generated by the relations
z2 + y2 ,
zy + y2 ,
zx + yx ,
y2v + x2 .

The Hilbert series for the cohomology ring is
-1 / t4 -2t3+ 2t -1.
Its denominator factors as ( t-1 )3 ( t+1 ) .

The Krull dimension of the cohomology ring is 3.
The longest regular sequence consists of the generators w , v .
A homogeneous set of parameters is the set w , v , z2 of degrees [ 2, 2, 2 ] .

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [] [ z+ y ]
ROW (0) : [1] [ y, z ] [ x ] [ yx ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.1 * G.2 * G.4, G.4, G.3 * G.4 ] The cohomology ring of E is a polynomial ring in the variables z , y , x .

The images of the generators of the cohomology of G restricted to E are
z ,
z ,
z2 + zy ,
z2 + zy + y2 + zx + x2 ,
z2 + y2
in the cohomology of E.

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

This ideal is also the nilradical of the cohomology of G.

It is nilpotent of degree 2.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

The group H is abelian of type [ 4, 2 ]

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

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

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

Maximal Subgroup H #2

The group H is abelian of type [ 4, 2 ]

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

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

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

Maximal Subgroup H #3

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

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

The images of the generators of the cohomology of G restricted to H are
z ,
z ,
z2 + zx ,
z2 + zy + y2 ,
z2 + x2
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 Isomorphic to the Group of Order 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3

of type Dihedral(8) .

The generators of G have images [ pcy.2, pcy.1 * pcy.2 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
z ,
z + y ,
y2 + w
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
y2 .

Maximal Quotient Group Q #2

The kernel of the quotient is generated by G.4 .

The group Q is abelian of type [ 4, 2 ] .

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

The images of the generators of the cohomology of Q inflated to G are
z + y ,
y ,
y2 + v
in the cohomology of G.

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

Maximal Quotient Group Q #3

The kernel of the quotient is generated by G.3 * G.4 .

The Group Q is Isomorphic to the Group of Order 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3

of type Dihedral(8) .

The generators of G have images [ pcy.1 * pcy.2, pcy.2 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
y ,
z + y ,
y2 + x + w + v
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
y2 .

Action of Automorphisms

The groups of outer automorphisms of G has order 8, and is generated by 3 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
z ,
y ,
x ,
w ,
v .

Automorphism #2

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
z ,
y ,
y2 + x ,
w ,
y2 + v .

Automorphism #3

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 ,
x + w + v ,
v .

CHECKS

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

THE COMPUTATION IS COMPLETE