Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 10

The explicit matrices areDirect matrix multiplication givesThe reverse products have the opposite signs: $YX=\hat{Z}$, $ZY=\hat{X}$, $XZ=\hat{Y}$. Since $W$ is the identity matrix and $X^2=Y^2=Z^2=-W$,These are the usual quaternion multiplication relations in the complex matrix representation of quaternions.

The products in (a), together with $(-A)B=A(-B)=-(AB)$ and $(-A)(-B)=AB$, give closure of the eight-element set. Associativity is inherited from matrix multiplication, $W$ is the identity element, and the inverses from (a), with $(-A{)}^{-1}=-A^{-1}$, remain in the set. Thus the group axioms hold. This is the quaternion group $Q_8$.

In the element order specified in the question, the multiplication table isInterpreting “character of the group” as the character of a representation specified by these matrices, take the trace of each matrix:A character belongs to a representation; the underlying abstract group can have other characters.

The three order-four subgroups areIn each case the generator has powers $A,A^2=\hat{W},A^3=\hat{A},A^4=W$, so all three are cyclic and mutually isomorphic to $C_4$, a cyclic group of order four. For $G_1$ the table isThe maps $X^j\mapsto Y^j$ and $X^j\mapsto Z^j$ are explicit group isomorphisms. These are all order-four subgroups: any such subgroup contains an element other than $W,\hat{W}$, and that element has order four and already generates the subgroup. They share the order-two subgroup $\{W,\hat{W}\}$ and are each normal, being of index two.

For $G_1=⟨X⟩$, define the one-dimensional complex group representationExplicitly $W\mapsto 1$, $X\mapsto i$, $\hat{W}\mapsto -1$ and $\hat{X}\mapsto -i$. Since $i^4=1$, the map is well-defined on exponents modulo four and preserves products. Its four images are distinct, so it is also a faithful representation. The same construction applies to either of the other cyclic subgroups.