True or False? ===help===?

Question

1. The Z4 group has 4 generators.
2. Z4 = <3>
3. The operation of a cyclic group is commutative
4. The operation of a group is always commutative
5. If n/m<1, then n 6. A proposition is a theorem
7. A square matrix has always a multiplicative inverse.
8. A square matrix has always an additive inverse.
9. A subgroup is a subset of a group.
10. A group has at least two subgroups.

Answer 1

1: False. Z4 has four elements, of which only 3 are generators (0 ∈ Z4, but <0> ≠ Z4)
2: True. The multiples of 3 in Z4 are {3, 2, 1, 0}, which is clearly every element in Z4
3: True. Every cyclic group Zn is isomorphic to the group of integers under addition mod n, whose operation is clearly commutative.
4: False. See, for instance, the group of invertible n×n matrices under multiplication.
5: False. Let n be any positve number and m be any negative number. Clearly, n/m <1 and m 6: True.
7: False. See any n×n matrix of rank strictly less than n.
8: True - the additive inverse is given by scalar multiplication by -1.
9: True.
10: False. The trivial group, consisting of only the identity element, has only one subgroup (specifically, itself).

Edited to add: just in case you're wondering about the previous answer - the empty set does not form a group, since it fails to contain an identity element.

Answer 2

1. The Z4 group has 4 generators.
F, only 2
2. Z4 = <3>
T
3. The operation of a cyclic group is commutative
T
4. The operation of a group is always commutative
F
5. If n/m<1, then n F
6. A proposition is a theorem
T
7. A square matrix has always a multiplicative inverse.
F
8. A square matrix has always an additive inverse.
T
9. A subgroup is a subset of a group.
F
10. A group has at least two subgroups
T

Answer 3

Here e is the identity of the group.

1) False, the identity is not a generator, since ={e}.
2) True. 3+3=2, 2+3=1 1+3=0. (As an additive group.)
3) True
4) False, Sn and Dn are not commutative
5) Only if n,m>0 Consider n=5 m=-3
6) True. A proposition is usually an easy theorem whose proof does not require any interesting techniques.
7) False The 2x2 matrix of all 0's does not have an inverse.
8) True, just negate all entries
9) True
10) For any non-trivial group: True, {e} and the group itself. If you mean proper subgroups it is false. Consider Zp where p is prime. The only proper subgroup is {e}.
Of course the trivial group has only one subgroup.

Answer 4

i don't know all the answers, well not off the top of my head, here are a few.

5. False. take n negative and m positive for counterexamples.
7. False. It will only have a mult.inverse if determinant is non-zero, i.e. it is non-singular.
8. True, just negate each element of the matrix to find additive inverse.
9. True.
10. EDIT
hopefully these ones are correct.

thanks pascal