State Lagrange’s theorem. What are the possible orders of subgroups of a group of order 12?
Lagrange’s theorem in algebra states that for any finite group $G$, the order (number of elements) of every subgroup $H$ of $G$ divides the order of the _____ _________ __________ _________ ____ ________ _________.
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Find the first term of the series whose second and subsequent terms are ,0,3,8 − 0,1 using difference table.
From the following data, calculate the population in the year 1985:
| Year | Population (in , 000) |
| 1971 | 12 |
| 1981 | 15 |
| 1991 | 20 |
| 2001 | 27 |
| 2011 | 49 |
Find the gcd of the polynomials x4 + 3x + 2 and x³ + 3x² + 5x + 3.
How many terms n be chosen in Maclaurin’s expansion for ex with an error less than
Given the data:
f(3) = 168,
f(7) = 120
and f(9) = 72
If f(k) is estimated as 138 using the Newton’s form of the interpolating polynomial, then find the value of .k
Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.
(a) Every subgroup of S3 is normal.
(b) Every abelian group is cyclic.
(c) In a ring with unity, the sum of any two units is a unit.
(d) If a field has characteristic p, p a prime, the field is finite.
(e) If every element in group has finite order, the group is finite.
Solve the system of equations:
x + 2y + z = 3
3x − 2y − z4 = −2
2x + 3y − z = −6
using Gauss-Elimination method.