Assertion: abelian group's subgroup | /r/HasProperty | whole group
Facets:
value: for an even stronger constraint
type: OTHER
value: informally
type: MANNER
Source sentences:
text: The group has no proper nontrivial subgroups: the only subgroups are the whole group and the trivial subgroup.
text: A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in
text: Informally, a subgroup is a group H contained within a bigger one, G. Concretely, the identity element of G is contained in H, and whenever h 1 and h 2 are in H, then so are h 1 ⋅ h 2
text: A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.[s]
text: In other words, a subgroup N of the group G is normal in G if and only if gng−1 ∈ N for all g ∈ G and n ∈ N.
text: Every group of prime order is cyclic, because Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group.
text: Since any subgroup-defining function (such as the center, the derived subgroup, the Frattini subgroup, the Fitting subgroup etc.) returns a characteristic subgroup of the whole group, and since every characteristic subgroup is normal, any subgroup obtained via a subgroup-defining function must be either trivial or the whole group.
text: The derived subgroup must be the whole group -- in other words, the group is perfect
text: A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N.
text: A subgroup N of a group G is a normal subgroup of G if and only if for all elements g of G the corresponding left and right cosets are equal, that is, gN = Ng.
text: For an even stronger constraint, a fully characteristic subgroup (also, fully invariant subgroup; cf. invariant subgroup), H, of a group G, is a group remaining invariant under every endomorphism of G; that is,
text: If e is the identity of G, then the trivial group {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself.