GROUP OF ORDER 16 #14

GROUP #14

Quaternion(16)

The MAGMA library number for G is 9

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

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

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x ] in degrees [ 1, 1, 4 ] , by the ideal generated by the relations
zy + y2 ,
z3 .

The Hilbert series for the cohomology ring is
-t2 -t -1 / t3 -t2+ t -1.
Its numerator factors as ( t2+t+1 ) .
Its denominator factors as ( t-1 ) ( t2+1 ) .

The Krull dimension of the cohomology ring is 1.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
x

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.4 ] The cohomology ring of E is a polynomial ring in the variables z

The images of the generators of the cohomology of G restricted to E are
0 ,
0 ,
z4
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 4.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

The group H is abelian of type [ 8 ]

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
0 ,
z ,
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 Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3 Generated by [ G.3, G.1 ]

of type Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
y ,
0 ,
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 #3

The Group H is Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3 Generated by [ G.3, G.1 * G.2 ]

of type Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
y ,
y ,
x
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 generated as an ideal by
y3 .

It is nilpotent of degree 2.
The annihilator of the Essential Cohomology has dimension 1 .
The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by x
The essential cohomology is generated as a module over Q by the elements
[] [] [ y3 ]

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1

The kernel of the quotient is generated by 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 ,
z2
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
z2 + y2 + x ,
zx + yx .

Action of Automorphisms

The groups of outer automorphisms of G has order 4, and is generated by 2 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

    G.1 * G.4
    G.2 * G.4
    G.3
    G.4 .

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

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

    G.1 * G.2 * G.3 * G.4
    G.2 * G.3
    G.3 * G.4
    G.4 .

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

CHECKS

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

THE COMPUTATION IS COMPLETE