Search results
Results from the WOW.Com Content Network
Likewise, itself is always a normal subgroup of (if these are the only normal subgroups, then is said to be simple). [6] Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup [ G , G ] {\displaystyle [G,G]} .
The intersection of subgroups A and B of G is again a subgroup of G. [5] For example, the intersection of the x-axis and y-axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.
The set of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal. Similarly, the additive group of the integers (, +) is not simple; the set of even integers is a non-trivial proper normal subgroup. [1]
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. A subgroup series is used in the subgroup method. Subgroup series are a special example of the use of filtrations in abstract algebra.
The centralizer and normalizer of S are both subgroups of G. Clearly, C G (S) ⊆ N G (S). In fact, C G (S) is always a normal subgroup of N G (S), being the kernel of the homomorphism N G (S) → Bij(S) and the group N G (S)/C G (S) acts by conjugation as a group of bijections on S.
Here are several examples: Let H be a nontrivial group, and let G be the direct product, H × H. Then the subgroups, {1} × H and H × {1}, are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism, (x, y) → (y, x), that switches the two factors.
There is a bijective correspondence between the subgroups of that contain and the subgroups of / ; if is a subgroup of containing , then the corresponding subgroup of / is . This correspondence holds for normal subgroups of G {\displaystyle G} and G / N {\displaystyle G\,/\,N} as well, and is formalized in the lattice theorem .
There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class: E p (G) is the intersection of all index p normal subgroups; G/E p (G) is an elementary abelian group, and is the largest elementary abelian p-group onto which G surjects.