group theory

1234567891011121314151617181920212223242526272829303132333435363738394041
Across
  1. 1. Two elements x and y are ___ if gxg^-1 = y for some element in the group.
  2. 3. A cube and an octahedron are these types of solids.
  3. 6. Known as a partial converse to Lagrange's theorem; a subgroup with an order that is a prime divisor of the order of the original group always exists.
  4. 8. The subgroup of a group is denoted by this symbol.
  5. 9. A type of group that describes the symmetries of a polygon.
  6. 10. The field of mathematics that is dealing with groups.
  7. 11. A property of two groups that are one-to-one and have the same multiplication operation.
  8. 13. A division of a set into disjoint sets whose union is the entire set.
  9. 16. A type of commutative group.
  10. 18. The set of all numbers denoted by a capital Z.
  11. 22. The theory that connects field theory and group theory.
  12. 24. This is the set of all images of an element in the set as g varies through the original group to produce another set called the orbit of x.
  13. 25. All groups are isomorphic to a subgroup of some permutation group.
  14. 26. A subset of the original set that forms a group under the same multiplication operation.
  15. 27. The five solids that are considered to be "regular".
  16. 29. This is a homomorphism from a group to the group of permutations of a set.
  17. 31. A transformation that keeps the original shape unchanged.
  18. 32. The rotational symmetry of a cube is isomorphic to that of S#___ where S is the permutation group of the answer.
  19. 34. A function that maps one group to another group that is isomorphic to it.
  20. 35. The number of symmetries of a tetrahedron.
  21. 37. The order of any subgroup is a divisor of the order of the original group.
  22. 38. A group consisting of the integers modulo n exists only if n obeys this property.
  23. 39. A group with elements consisting of (0, 0), (0, 1), (1, 0), (1, 1).
  24. 40. A set with a multiplication operation.
  25. 41. This is similar to an isomorphism except it does not have to be a bijection.
Down
  1. 2. One of the three group axioms.
  2. 4. The dual solid of an icosahedron.
  3. 5. The Lorentz group is one example of this.
  4. 7. A bijection from one set to itself.
  5. 12. This is the group of all g that leave x fixed if x is a part of the set.
  6. 14. This is a subgroup of the permutation group.
  7. 15. A one-to-one correspondence from one set onto another.
  8. 17. The group G contains at least one subgroup of order p^m where p is a prime number that divides |G| and m is the highest power of p such that p^m divides |G|.
  9. 19. This is the number of group axioms that there are.
  10. 20. Any two subgroups that have an order of p^m are conjugate.
  11. 21. This is the set of all elements that are mapped to the identity element in the new group.
  12. 23. The rotational symmetry group of a dodecahedron is isomorphic to that of the A#___ where A denotes the alternating group of this number.
  13. 28. The basic accepted truths that form a theory.
  14. 30. The number of subgroups that have an order of p^m is congruent to one modulo p and is also a factor of |G| / p^m.
  15. 33. The "superset" of group theory and also ring theory.
  16. 36. A group in special relativity that represents the set of all possible transformations of spacetime.