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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.
The character theory of Z-groups is well understood (Çelik 1976), as they are monomial groups. The derived length of a Z-group is at most 2, so Z-groups may be insufficient for some uses. A generalization due to Hall are the A-groups, those groups with abelian Sylow subgroups.
The dihedral group Dih 4 has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same cyclic subgroup of order four. In addition, there are two subgroups of the form Z 2 × Z 2, generated by pairs of order-two ...
Specifically, all subgroups of Z are of the form m = mZ, with m a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0Z, they all are isomorphic to Z. The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility. [10]
There is a natural homomorphism SL(2, Z) → SL(2, Z/NZ) given by reducing the entries modulo N. This induces a homomorphism on the modular group PSL(2, Z) → PSL(2, Z/NZ). The kernel of this homomorphism is called the principal congruence subgroup of level N, denoted Γ(N). We have the following short exact sequence:
The union of subgroups A and B is a subgroup if and only if A ⊆ B or B ⊆ A. A non-example: 2 Z ∪ 3 Z {\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} } is not a subgroup of Z , {\displaystyle \mathbb {Z} ,} because 2 and 3 are elements of this subset whose sum, 5, is not in the subset.
In the case of n = 2 this gives the rather obvious result that a subgroup H of index 2 is a normal subgroup, because the normal subgroup of H must have index 2 in G and therefore be identical to H. (We can arrive at this fact also by noting that all the elements of G that are not in H constitute the right coset of H and also the left coset, so ...
In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, {1, −1}, is characteristic, since it is the only subgroup of order 2. If n > 2 is even, the dihedral group of order 2n has 3 subgroups of index 2, all of which are normal. One of these is the ...