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.

- 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

- Let
**Z**be the group of integers. Which of the following are subgroups? If possible also write it as a principal ideal:- {0},
- {
`n`∈**Z**:`n`≥0 }, - {
`n`∈**Z**:`n`even }, - {
`n`∈**Z**:`n`divisible by 6 and by 9 }.

- In the ring
`R`=**Z**/(1000), consider the ideal generated by 30,`I`=30`R`. Determine the size of the set`I`\ 16`R`.

- 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?- All polynomials whose coefficient of
`x`is zero; - All polynomials whose degree is even together with the zero polynomial;
- All polynomials with integer coefficients.

- All polynomials whose coefficient of
- In a cyclic group of order 15 consider an element
`x`such that the set {`x`^{3},`x`^{5},`x`^{9}} has exactly 2 elements. How many elements does {`x`^{n}:`n`∈**Z**,`n`≥0 } have? - If
`A`is a unital (non-necessarily commutative) associative**Z**-algebra and for each`a∈ A`we have`a`^{2}=`a`. Which of the following must be true:`a + a`= 0 for every`a ∈ A`;`(a+b)`for every^{2}= a^{2}+ b^{2}`a,b ∈ A`;`A`is commutative.

- What is the greatest integer that divides
`p`^{4}-1 for every prime`p`∈**P**.- 12,
- 30,
- 48,
- 120,
- 240.

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