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A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). [2] [3] If H is a subgroup of G, then G is sometimes called an overgroup of H.
Since the normal subgroup is a subgroup of H, its index in G must be n times its index inside H. Its index in G must also correspond to a subgroup of the symmetric group S n, the group of permutations of n objects. So for example if n is 5, the index cannot be 15 even though this divides 5!, because there is no subgroup of order 15 in S 5.
One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2. Order p 4: The classification is complicated, and gets much harder as the exponent of p increases.
A core-free subgroup is a subgroup whose normal core is the trivial subgroup. Equivalently, it is a subgroup that occurs as the isotropy subgroup of a transitive, faithful group action. The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words, H is a maximal element of the partially ordered set of subgroups of G that are not equal to G.
In geometric group theory, a discipline of mathematics, subgroup distortion measures the extent to which an overgroup can reduce the complexity of a group's word problem. [1] Like much of geometric group theory, the concept is due to Misha Gromov , who introduced it in 1993.
A subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup. ∀φ ∈ Inn(G)： φ(H) ≤ H. Since Inn(G) ⊆ Aut(G) and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic.
In mathematics, specifically group theory, a subgroup series of a group is a chain of subgroups: = = where is the trivial subgroup.Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups.