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20 Questions Submissions

You can find the 20 Questions in the 20Questions folder in the 'Files" section of the Canvas site.

Some general remarks about some of the questions I saw: 

  1. “Is the group a quotient group?” — Any group is a quotient group! (Why?)
  2. “Are the elements of the group integers/rational numbers/real numbers?” — These are great question to pin down what the *set* underlying the group is, but may not help with pinning down the group up to isomorphism. For instance, for any n>0, Z/nZ can be embedded as a subgroup of C-{0} under multiplication (why?). 
  3. Related to the above: “Do the elements consist of rotations of a physical object?” Z/nZ is isomorphic to the group of rotations of an n-gon; but this question will again only help you pin down the “underlying set” of the group, rather than its isomorphism type. A different question could have been: “Is the group a subgroup of the group of rotations of some object?”
  4. “Is your group operation addition?” Is along similar lines—one could have thought of the Nth roots of unity on the circle, under multiplication; but this is isomorphic to Z/NZ under addition. So perhaps “Are you abelian” would have been a more helpful question, or “Is there a common group isomorphic to your group where the operation is addition” would also have been more helpful.

Some great questions:

  1. Does the group act (insert adverbial phrase) on (insert set). (e.g., act without fixed points, or act transitively, on a set with K elements, or on R^n.)
  2. Does the group have any elements of order (insert number)? Also, does the group have K elements of order N?
  3. Does the group have K subgroups of order N?
  4. Does the group have K subgroups?
  5. Is the group isomorphic to a product of two non-trivial groups?
  6. Does the group have any non-trivial (i.e., not {e} and not G itself) normal subgroups?
  7. Is it abelian?
  8. Is it cyclic?
  9. Did we cover this group in class?
  10. “What gives you the right?” (after learning the group is of order 324.)

Some of the groups you thought of:

  1. S^1 (which is actually isomorphic to S^1 x R^n for any n > 0)
  2. S^1 x Z/2Z
  3. C under addition (which is actually isomorphic to R^n for any n>0).
  4. R (which is actually isomorphic to R^n for any n > 0)
  5. R/Z (which is isomorphic to S^1)
  6. R/Q
  7. Z
  8. Z/1001Z
  9. Z/3Z x Z/3Z x Z/3Z x Z/3Z x Z/2Z x Z/2Z
  10. Z/2Z x Z/2Z x Z/2Z
  11. Z/2Z x Z/4Z
  12. Z/8Z
  13. (Q-0, x)
  14. Q_8
  15. D_12
  16. D_24
  17. D_8
  18. S_4
  19. A_4
  20. Free group on two generators
  21. Free group on three generators
  22. Free abelian group on three generators, subject to the relation x=7y (though we haven’t talked about free abelian groups yet)
  23. O(3)
  24. GL_3(R)
  25. GL_2(C)
  26. GL_2(R)
  27. SL_2(R)
  28. The group of all power series where the constant term is not 0, under multiplication of power series.