An example of a quaternionic representation would be the four-dimensional real irreducible representation of the quaternion group " Q " 8.
22.
P . R . Girard's essay " The quaternion group and modern physics " discusses some roles of quaternions in physics.
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"' Sign representations with i, j, k-kernel "': the quaternion group has three maximal normal subgroups : the cyclic subgroups generated by respectively.
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The quaternion group has the unusual property of being Hamiltonian : every subgroup of Q is a normal subgroup, but the group is non-abelian.
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Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional unit or a zero divisor.
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The character table does not in general determine the group up to isomorphism : for example, the quaternion group and the dihedral group of elements,, have the same character table.
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The trick is just going to be when we put the three copies of the field together, do we do it like the dihedral group or like the quaternion group.
28.
Together the dihedral, semidihedral, and quaternion groups form the 2-groups of maximal class, that is those groups of order 2 " n " + 1 and nilpotency class " n ".
29.
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.
30.
This same paper gives a number of examples of groups which cannot be realized as Zappa Sz�p products of proper subgroups, such as the quaternion group and the alternating group of degree 6.
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