No two of D₆, A₄, and T are isomorphic, where T is the group of order 12 described in Proposition 6.4 and Exercise 5.

Data from in exercise 5

(a) Show that there is a nonabelian subgroup T of S₃ X Z₄ of order 12 generated by elements a,b such that |a| = 6, a³ = b², ba = a^-1b.(b) Any group of order 12 with generators a,b such that lal = 6, a³ = b², ba = a^-1b is isomorphic to T.

Transcribed Image Text:

Proposition 6.4. There are (up to isomorphism) exactly three distinct nonabelian groups of order 12: the dihedral group De, the alternating group A, and a group T generated by elements a,b such that |a| = 6, b² = a³, and ba = a-¹b.