Abstract
Frattini subgroup, Φ(G), of a group G is the intersection of all the maximal subgroups of G, or else G itself if G has no maximal subgroups. If G is a p-group, then Φ(G) is the smallest normal subgroup N such the quotient group G/N is an elementary abelian group. It is against this background that the concept of p-subgroup and fitting subgroup play a significant role in determining Frattini subgroup (especially its order) of dihedral groups. A lot of scholars have written on Frattini subgroup, but no substantial relationship has so far been identified between the parent group G and its Frattini subgroup Φ(G) which this tries to establish using the approach of Jelten B. Napthali who determined some internal properties of non abelian groups where the centre Z(G) takes its maximum size.
Highlights
Giovani Frattini (1852–1925), introduced another type of subgroup known as Frattini subgroup denoted by Φ(G)
He described the elements of a finite group into two classes; generators and non-generators and noticed that the nongenerators form normal subgroups, called the Frattini subgroup, which equals the intersection of all maximal subgroups of the given group
James and Paul (2017) who studied the relations between nilpotency and Frattini subgroups, remarked that if a maximal subgroup M of a group G is normal, G/M has prime order and G ′ is a subgroup of M if and only if G ′ is a subgroup of Φ(G)
Summary
Giovani Frattini (1852–1925), introduced another type of subgroup known as Frattini subgroup denoted by Φ(G) He described the elements of a finite group into two classes; generators and non-generators and noticed that the nongenerators form normal subgroups, called the Frattini subgroup, which equals the intersection of all maximal subgroups of the given group. James and Paul (2017): who studied the relations between nilpotency and Frattini subgroups, devoted their work to the structure of finite nilpotent algebras. They remarked that if a maximal subgroup M of a group G is normal, G/M has prime order and G ′ is a subgroup of M. Definition: The order of Frattini subgroup of any dihedral group whose order is a power of two is exactly (2k ), k ≥ 3
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