The group defined by generators a, b and relations aⁿ = e (3 ≤ n ϵ N*), b² = e and abab = e is the dihedral group D_n. [See Theorem 6.13.]

Theorem 6.13

For each n ≥ 3 the dihedral group D_n is a group of order 2n whose generarors a and b sarisfy:
(i) aⁿ = (1); b² = (1); a^k ≠ (1) if O < k < n;
(ii) ba = a^-1b.

Any group G which is generated by elements a, b ϵ G satisfying (i) and (ii) for some n ≥ 3 (with e ϵ G in place of(1)) is isomorphic to D_n.

SKETCH OF PROOF.

Verify that a,b ϵ D,. as defined above satisfy (i) and (ii), whence D_n = (a, b) = { aⁱbⁱ| 0 ≤ i < n; j = 0,1} (see Theorem 2.8). Then verify that the 2n elements aⁱbⁱ (0 ≤ i < n;j = 0,1) are all distinct (just check their action on 1 and 2), whence |D_n| = 2n. Suppose G is a group generated by a,b ϵ G and a,b satisfy (i) and (ii) for some n ≥ 3. By Theorem 2.8 every element of G is a finite product a^m1b^m2a^m3b^m4, • • • b^mk (m_i ϵ Z). By repeated use of (i) and (ii) any such product may be written in the form aⁱbⁱ with 0 ≤ i < n and j = 0,1 (in particular note that b² = e and (ii) imply b = b^-1 and ab= ba^-1). Denote the generators of D_n by a₁,b₁ to avoid confusion and verify that the map ∫: D_n → G given by a₁ⁱb₁ⁱ → aⁱbⁱ is an epimorphism of groups. To complete the proof we show that ∫ is a monomorphism. Suppose ∫ (a₁ⁱb₁ⁱ) = aⁱbⁱ = e ϵ G with 0 ≤ i < n and j = 0,1. If j = 1, then aⁱ = b. and by (ii) aⁱ⁺¹ = aⁱa = ba = a^-1b = a^-1aⁱ = a^i-1, which implies a² = e. This contradicts (i) since n ≥ 3. Therefore j = 0 and e = aⁱb^o = aⁱ with 0 ≤ i < n, which implies i = 0 by (i). Thus ∫(a₁ⁱb₁ⁱ) = e implies a₁ⁱb₁ⁱ = a₁^°b₁^° = (1). Therefore ∫ is a monomorphism by Theorem 2.3.
This theorem is an example of a characterization of a group in terms of "generators and relations." A detailed discussion of this idea will be given in Section 9.