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List of all nonabelian groups up to order 31 Order Id. [a] G o i Group Non-trivial proper subgroups [1] Cycle graph Properties 6 7 G 6 1: D 6 = S 3 = Z 3 ⋊ Z 2: Z 3, Z 2 (3) : Dihedral group, Dih 3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
More generally, a quasisimple group (a perfect central extension of a simple group) that is a non-trivial extension (and therefore not a simple group itself) is perfect but not simple; this includes all the insoluble non-simple finite special linear groups SL(n,q) as extensions of the projective special linear group PSL(n,q) (SL(2,5) is an ...
A non-finitely generated countable example is given by the additive group of the polynomial ring [] (the free abelian group of countable rank). More complicated examples are the additive group of the rational field Q {\displaystyle \mathbb {Q} } , or its subgroups such as Z [ p − 1 ] {\displaystyle \mathbb {Z} [p^{-1}]} (rational numbers ...
Every abelian group is a CA-group, and a group with a non-trivial center is a CA-group if and only if it is abelian. The finite CA-groups are classified: the solvable ones are semidirect products of abelian groups by cyclic groups such that every non-trivial element acts fixed-point-freely and include groups such as the dihedral groups of order 4k+2, and the alternating group on 4 points of ...
This is in fact the significance in group-theoretical terms of the unique non-trivial element of (/,),. An example of a second cohomology group is the Brauer group: it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure:
The dihedral group D 8 of order 8, which can also be given by either of the two constructions in the section above for p = 2 (for p odd they give different groups, but for p = 2 they give the same group). This group has 2 elements of order 4. The quaternion group Q 8 of order 8, which has 6 elements of order 4. If n is a positive integer there ...
A group G is called the direct sum [1] [2] of two subgroups H 1 and H 2 if each H 1 and H 2 are normal subgroups of G, the subgroups H 1 and H 2 have trivial intersection (i.e., having only the identity element of G in common), G = H 1, H 2 ; in other words, G is generated by the subgroups H 1 and H 2.
F 4 (q) has a non-trivial graph automorphism when q is a power of 2. These groups are the automorphism groups of 8-dimensional Cayley algebras over finite fields, which gives them 7-dimensional representations. They also act on the corresponding Lie algebras of dimension 14. G 2 (q) has a non-trivial graph automorphism when q is a power of 3