1.  Sn is non-Abelian for all n >= 3. 

2.  If a permutation that is an m-cycle and b is

    a permutation thet is an n-cycle, then |ab| = lcm(m,n).

3.  Every group is isomorphic to a group of permutations.

4.  Every infinite cyclic group is isomorphic to Z.

5.  Every cyclic group of order n is isomorphic to Zn

6.  Dn has a subgroup isomorphic to Zn.

7.  A group can be isomorphic to a proper subgroup of itself.

8.  Two groups isomorphic to the same group are isomorphic to each other.

9.  If a finite group has order n, then the group contains a subgroup

    of order d for every positive divisor d of n.

10.  If a belongs to a finite group, then |a| divides |G|.

11.  If H is a subgroup of G and a belongs to G, then |aH|=|Ha|.

12.  Us(st) is isomorphic to U(t).

13.  The external direct product of cyclic groups is cyclic.

14.  The external direct product of D3 and D4 is isomorphic to D12

15.  U(st) is isomorphic to the external direct product of U(s) and U(t).

16.  For Groups G and H, |G+H| = |G||H|.

17.  If (a,b) belongs to G+H, THen |a,b| = lcm (|a|,|b|).

18.  If m is odd then U(2m) is isomorphic to U(m).

19.  If p is a prime then U(p^n) is cyclic.

20.  If G and H are finite groups that have the same number of each

     order then G and H are isomorphic.

   
1.  Any group has only one element of order 1.

2.  In a group (ab)&(-1) = a^(-1) b^(-1).

3.  In general, you cannot take roots in groups.

4.  You cannot divide in groups.

5.  The union of two subgroups is a subgroup.

6.  The intersection of two subgroups is a subgroup.

7.  Every group is a subgroup of itself.

8.  A finite non-empty subset of a group that is closed is a subgroup.

9.  For every positive integer n there is a cyclic group of order n.

10. Every finite cyclic group contains an element of every order
 
    that divides the order of the group.

11. Every cyclic group is abelian.

12. Every element of a group generates a cyclic subgroup of the group.

13. In general, there is no relationship between |ab| and |a| and |b|.

14. If |a| and |b| are finite, then |ab| is finite.

15.  Every cyclic group of order at least 3 has at least two generators.

16.  If a and b are elements of an Abelian group, then
 
     |ab| is the lcm ( |a|,  |b|).

17.  If g is a group element and g^n = e, then |g| = n.

18.  The rotations of a dihedral group form a cyclic subgroup.

19.  |Dn| = n.

20.  If a group has an element of order 15 it must have at

     least 8 elements of order 15.

21.  If a group has more than 8 elements of order 15, it must have

     at least 16 elements of order 15.

22.  If a group has an element of order 10, then the number of

     elements of order 10 is divisible by 4.

23.  An is a subgroup of Z.

24.  A subgroup of a cyclic group is cyclic.

25.  A subgroup of an Abelian group is Abelian.

26.  C(a), the centralizer of a, is an Abelian group.

27.  In a finite cyclic group, the number of elements of order d 

     is phi(d).