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
    abcd
    aabcd
    bbcda
    ccdab
    ddabc
    II
    abcd
    aabcd
    bbadc
    ccdaa
    ddcab
    III
    *abcd
    aabcd
    bbadc
    ccdcd
    ddcdc

  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.

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