Equivalence of definitions of maximal subgroup of group of prime power order

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This article gives a proof/explanation of the equivalence of multiple definitions for the term maximal subgroup of group of prime power order
View a complete list of pages giving proofs of equivalence of definitions

Statement

Suppose G is a group of prime power order and H is a subgroup of G. The following are equivalent:

  1. H is a maximal subgroup of G.
  2. H is a maximal normal subgroup of G.
  3. H is a normal maximal subgroup of G.
  4. H is a subgroup of prime index in G.
  5. H is a normal subgroup of G and the quotient group G/H is a group of prime power order.
  6. H contains the Frattini subgroup \Phi(G) of G and the quotient H/\Phi(G) is a codimension one subspace in the Frattini quotient G/\Phi(G), viewed as a vector space over the field of p elements.

Facts used

  1. Prime power order implies nilpotent
  2. Nilpotent implies every maximal subgroup is normal
  3. Equivalence of definitions of group of prime power order