Characterizing Abelian Minimal Normal Subgroups with Complements in Group Theory

In group theory, the concept of a complemented subgroup is fundamental. Specifically, if a subgroup (N) has a complement (H) in a group (G), this means there exists a subgroup (H leq G) such that (G AH) and (A cap H {1}). This article delves into the conditions under which an abelian minimal normal subgroup (N) of (G) is complemented, focusing on the role of the Frattini subgroup (Phi(G)).

Introduction to the Frattini Subgroup

The (Phi(G)) (Frattini subgroup) of a group (G) consists of all the non-generators of (G). An element (g in G) is a non-generator if whenever (G) is generated by (g) together with a set (S subset G), it is also true that (G) is generated by (S) without (g).

Conditions for Complemented Subgroups

The statement that an abelian minimal normal subgroup (N) is complemented in (G) if and only if (N cap Phi(G) {1}) is a critical condition in group theory. Here, we explore the necessity and sufficiency of this condition.

Necessity

If (N cap Phi(G) {1}), we need to show that (N) has a complement. We assume (N eq 1) and (N cap Phi(G) {1}). This implies there is a maximal subgroup (M) of (G) that does not contain (N). Therefore, (NM G).

By the normality of (N), (N cap M) is normal in both (N) since (N) is abelian, and (M) since (N lhd G). Hence, (N cap M lhd G). Since (N cap M) is a proper subgroup of (N), by induction, there is a complement (K) to (N cap M) in (G). To prove that (M cap K) is a complement to (N) in (G), we need to show that (N cap (M cap K) {1}).

Sufficiency

If (N) is an abelian minimal normal subgroup of (G) that is not contained in (Phi(G)), then (N) is complemented in (G). Since (N leq Phi(G)), there must be a maximal subgroup (H) of (G) that does not contain (N). The normality of (N) implies that (NH) is a subgroup of (G) containing (H). By the maximality of (H), (NH G).

Since (N) is abelian with (N cap H) as a subgroup, and (N) is a minimal normal subgroup, it follows that (N cap H {1}). Therefore, (N) is complemented in (G).

Counterexamples and Illustrations

Consider the quaternion group (Q_8 {pm 1, pm i, pm j, pm k}). It has three maximal subgroups: (N_i {pm 1, pm i}), (N_j {pm 1, pm j}), and (N_k {pm 1, pm k}), all of which are normal and abelian. The intersection of these is the Frattini subgroup (Phi(Q_8) {1, -1}), which is also the center. Here, (N_i) is a normal abelian subgroup of (Q_8) that is not contained in (Phi(Q_8)), but (N_i) is not complemented in (Q_8) because if it were, the complement would have to be of order 2, specifically ({1, -1}). However, (Q_8 eq N_i {1, -1}).

Conclusion

In conclusion, the characterization of abelian minimal normal subgroups that are complemented in a group is a nuanced topic requiring a deep understanding of the Frattini subgroup and the specific conditions under which such complements exist. Understanding these concepts is crucial for advanced work in group theory and can serve as a foundation for further exploration into the structure of groups and their subgroups.