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Subgroup Invariant Under Conjugation

Introduction

In the captivating realm of abstract algebra , a normal subgroup is a hidden gem - a subgroup that remains steadfast, unmoved by the whirlwind of conjugation within the encompassing group . These unassuming heroes are the building blocks that allow us to construct the enigmatic quotient groups , unlocking new avenues of exploration.

Évariste Galois, the legendary mathematician, was the first to grasp the profound significance of normal subgroups, paving the way for a deeper understanding of group theory. [2] These special subgroups are the precise kernels of group homomorphisms , serving as a means to internally classify the intricate web of homomorphisms defined on a group.

Definitions and Equivalent Conditions

A subgroup N of a group G is called a normal subgroup if it is invariant under conjugation; that is, the conjugation of any element of N by any element of G always remains within N. [3] This relationship is denoted as N ⊴ G.

There are several equivalent conditions that define a normal subgroup:

1. The image of conjugation of N by any element of G is a subset of N. [4]
2. The image of conjugation of N by any element of G is equal to N. [4]
3. The left and right cosets of N in G are equal. [4]
4. The sets of left and right cosets of N in G coincide. [4]
5. Multiplication in G preserves the equivalence relation “is in the same left coset as”. [4]
6. There exists a group on the set of left cosets of N in G, where multiplication of any two left cosets yields the left coset of their product. [2]
7. N is a union of conjugacy classes of G. [2]
8. N is preserved by the inner automorphisms of G. [5]
9. There is some group homomorphism G → H whose kernel is N. [2]
10. There exists a group homomorphism φ : G → H whose fibers form a group where the identity element is N and multiplication of any two fibers yields the fiber of their product. [2]
11. There is some congruence relation on G for which the equivalence class of the identity element is N.
12. For all n ∈ N and g ∈ G, the commutator [n, g] is in N. [citation needed]
13. Any two elements commute modulo the normal subgroup membership relation. [citation needed]

These equivalent conditions highlight the multifaceted nature of normal subgroups and their deep connections to various algebraic structures and concepts.

Examples

For any group G, the trivial subgroup {e} consisting of just the identity element is always a normal subgroup of G. Likewise, G itself is always a normal subgroup of G. [6] Other named normal subgroups include the center of the group and the commutator subgroup [G, G]. [7] [8]

More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup. [9] If G is an abelian group , then every subgroup of G is normal. [10] More broadly, every subgroup of the center Z(G) of G is normal in G.

A concrete example of a normal subgroup is the subgroup N = {(1), (123), (132)} of the symmetric group S_3, consisting of the identity and both three-cycles. In contrast, the subgroup H = {(1), (12)} is not normal in S_3. [11] This illustrates the general fact that any subgroup of index two is normal.

As an example from matrix groups, consider the general linear group GL_n(R) of all invertible n×n matrices with real entries and its subgroup SL_n(R) of all n×n matrices of determinant 1 (the special linear group ). The subgroup SL_n(R) is normal in GL_n(R) because conjugating any matrix in SL_n(R) by an invertible matrix preserves the determinant being 1. [a]

In the Rubik’s Cube group , the subgroups consisting of operations that only affect the orientations of either the corner pieces or the edge pieces are normal. [12] The translation group is also a normal subgroup of the Euclidean group in any dimension. [13]

Properties

Normal subgroups exhibit several important properties:

1. If H is a normal subgroup of G, and K is a subgroup of G containing H, then H is a normal subgroup of K. [14]
2. Normality is not a transitive relation - a normal subgroup of a normal subgroup need not be normal in the original group. [15] However, a characteristic subgroup of a normal subgroup is always normal. [16] A group in which normality is transitive is called a T-group . [17]
3. The direct product of normal subgroups is also a normal subgroup. [20]
4. If G is a semidirect product G = N ⋊ H, then N is normal in G, though H need not be normal in G. [20]
5. If M and N are normal subgroups of an additive group G such that G = M + N and M ∩ N = {0}, then G = M ⊕ N. [18]
6. Normality is preserved under surjective homomorphisms and inverse images. [19]

Normal Subgroups, Quotient Groups, and Homomorphisms

The normal subgroups of a group G form a lattice under subset inclusion , with the least element being the trivial subgroup {e} and the greatest element being G itself. The meet (intersection) and join (product) of normal subgroups are also normal.

There is a deep connection between normal subgroups and group homomorphisms. If N is a normal subgroup of G, we can define a well-defined multiplication on the set of cosets G/N, turning it into a group called the quotient group of G modulo N. [23]

Furthermore, the kernel of any group homomorphism f : G → H is always a normal subgroup of G, and the image f(G) is isomorphic to the quotient group G/ker(f) (the first isomorphism theorem ). [24] In fact, the normal subgroups of G correspond bijectively (up to isomorphism) to the homomorphic images of G. [25] The normal subgroups are precisely the kernels of homomorphisms with domain G. [26]

Conclusion

Normal subgroups are fundamental objects in group theory, serving as the building blocks for quotient groups and the internal classification of group homomorphisms. Their rich properties and connections to other algebraic structures make them an essential topic for any student or researcher delving into the captivating world of abstract algebra.