GROUP OF ORDER 8 #5

GROUP #5

Quaternion(8)

The MAGMA library number for G is 4

GrpPC : G of order 8 = 2^3 PC-Relations: G.1^2 = G.3, G.2^2 = G.3, 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 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
z2 + zy + y2 ,
y3 .

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.3 ] 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 [ 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
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 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 ,
0 ,
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 ,
y2
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
zy ,
y2 .

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
[] [ zy, y2 ] [ zy2 ]

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
z2 + zy + y2 ,
y3 .

Action of Automorphisms

The groups of outer automorphisms of G has order 6, and is generated by 2 elements.

Automorphism #1

The order of the class of the automorphism in the outer automorphism group is 3. The images of the generators of G are

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
y ,
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.1 * G.2
    G.3 .

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

CHECKS

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

THE COMPUTATION IS COMPLETE