Group orders, element orders and spectra

Arithmetic of finite groups through orders: cyclic numbers and groups of a given order (groups of order 10, dihedral groups from an index-two cyclic subgroup), counts of elements of each order, the spectrum (set of element orders), recognition of groups by spectrum or by order and spectrum, Thompson's type, and totient sums; conjugacy-class sizes go to Conjugacy classes.

TheoremDihedral groups from a cyclic subgroup of index two#✓fully proved

Let \(G\) be a finite group and let \(H\) be a cyclic subgroup of \(G\) of index \(2\). If \(g^2 = 1\) for every element \(g\) of \(G\) that is not in \(H\), then \(G\) is isomorphic to the dihedral group \(D_n\) of order \(2n\), where \(n = |H|\).

theorem nonempty_mulEquiv_dihedralGroup_of_index_eq_two {G : Type*} [Group G] [Finite G]
    (H : Subgroup G) (hHcyc : IsCyclic H) (hindex : H.index = 2)
    (houtside : ∀ g : G, g ∉ H → g ^ 2 = 1) :
    Nonempty (G ≃* DihedralGroup (Nat.card H))

Used by 2

Used byDickson's classification, tame case · Noncyclic groups of order 10 are dihedral

• Counts of elements of each order
• Cyclic numbers
• Element orders of extensions of projective linear groups
• Finite groups with the same spectrum
• Groups of order 10
• Isospectral finite simple groups
• Number of groups with a given spectrum
• Order-divisibility bijections onto cyclic groups
• Prime divisors of element orders in soluble groups
• Recognition by order and spectrum
• Recognition of alternating groups by spectrum
• Recognition of PSL(2, 7) by spectrum
• Subgroups with all prime divisors of a simple group
• Thompson's problem on groups of the same type
• Totient sums of element orders

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