where e is the identity element and e commutes with the other elements of the group.
In the diagrams for D4, the group elements are marked with their action on a letter F in the defining representation R2. The same cannot be done for Q8, since it has no faithful representation in R2 or R3. D4 can be realized as a subset of the split-quaternions in the same way that Q8 can be viewed as a subset of the quaternions.
The quaternion group Q8 and the dihedral group D4 are the two smallest examples of a nilpotent non-abelian group.
Sign representations with i,j,k-kernel: Q8 has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup N, we obtain a one-dimensional representation factoring through the 2-element quotient group G/N. The representation sends elements of N to 1, and elements outside N to -1.
2-dimensional representation: Described below in Matrix representations.
Also, this representation permutes the eight non-zero vectors of (F3)2, giving an embedding of Q8 in the symmetric group S8, in addition to the embeddings given by the regular representations.
The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.
A generalized quaternion group Q4n of order 4n is defined by the presentation
The generalized quaternion groups have the property that every abelian subgroup is cyclic. It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group. In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL2(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, (Gorenstein 1980, p. 42). Letting pr be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL2(F) is 2n, where n = ord2(p2 − 1) + ord2(r).
The Brauer–Suzuki theorem shows that the groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.
Another terminology reserves the name "generalized quaternion group" for a dicyclic group of order a power of 2, which admits the presentation