Prime radical

The prime radical of a finite group G is the subgroup generated by all of its elements of prime order — equivalently, the join of all its minimal subgroups. It is a characteristic subgroup that captures the "bottom layer" of G without the normality restriction of the socle.

Definition

A minimal subgroup is a non-trivial subgroup with no proper non-trivial subgroup — necessarily cyclic of prime order. The prime radical is their join:

⟨ g ∈ G : |g| is prime ⟩ = ⟨ H : H a minimal subgroup of G ⟩.

For a p-group this is the omega subgroup Ω1(G) = ⟨ g : gp = e ⟩; in general it is the join of the Ω₁ of the Sylow subgroups.

Properties

• Characteristic. An automorphism preserves element orders, so it permutes the prime-order elements and fixes the subgroup they generate. In particular the prime radical is normal.
• Contains the socle. soc(G) ≤ prime radical(G): every minimal normal subgroup is generated by its prime-order elements. The two coincide when every minimal subgroup is normal — for instance whenever G is abelian, where both equal the elements of square-free order.
• Non-trivial. Every non-trivial finite group has an element of prime order, so the prime radical is > {e} (unlike a subgroup generated by a single class, it never collapses).
• Not the socle. Dropping "normal" can make it strictly larger: a group generated by its involutions — every symmetric group, every non-abelian simple group — has prime radical = G, even when its socle is a proper subgroup.

Examples

Q₈ is the sharp case: it has a unique element of order 2, so its prime radical is just that central C₂ — a proper subgroup. D₄, by contrast, has five involutions that together generate the whole group.

See also

• Socle — the join of the minimal normal subgroups; always contained in the prime radical.
• Cosocle — the lattice-dual of the socle.
• Subgroup — minimal subgroups are the atoms of the subgroup lattice.
• Wikipedia: Socle § Related