# GRE math, algebra sample questions (2000)

In the original GRE (Graduate Record Examinations) test you have about 2.5min per question (maybe a bit more for more difficult questions, but then also less time for easier questions). ©Educational Testing Service 2007.

1. Which of the following are tables of a group with four elements? In this case indicate the neutral element and identify with one of the standard groups
I
 ⋅ a b c d a a b c d b b c d a c c d a b d d a b c
II
 ○ a b c d a a b c d b b a d c c c d a a d d c a b
III
 * a b c d a a b c d b b a d c c c d c d d d c d c

2. Let Z be the group of integers. Which of the following are subgroups? If possible also write it as a principal ideal:
1. {0},
2. { nZ: n≥0 },
3. { nZ: n even },
4. { nZ: n divisible by 6 and by 9 }.

3. In the ring R = Z/(1000), consider the ideal generated by 30, I=30R. Determine the size of the set I \ 16R.
4. Let R be the field of real numbers and R[x] be the ring of polynomials in x over R. Which of the following are subrings?
1. All polynomials whose coefficient of x is zero;
2. All polynomials whose degree is even together with the zero polynomial;
3. All polynomials with integer coefficients.

5. In a cyclic group of order 15 consider an element x such that the set { x3, x5, x9 } has exactly 2 elements. How many elements does { xn: nZ, n ≥0 } have?
6. If A is a unital (non-necessarily commutative) associative Z-algebra and for each a∈ A we have a2 = a. Which of the following must be true:
1. a + a = 0 for every a ∈ A;
2. (a+b)2 = a2 + b2 for every a,b ∈ A;
3. A is commutative.

7. What is the greatest integer that divides p4-1 for every prime pP.
1. 12,
2. 30,
3. 48,
4. 120,
5. 240.