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.
(a) False. Not every subgroup of $S_3$ is normal. Consider the subgroup $H = \{e, (12)\}$. Conjugating by $(13) \in S_3$, we get $(13)(12)(13) = (23) \notin H$. Hence $H$ is ________ _________ _____ ____ _____.
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If F is a field, show that U(F[x]) = F.
Find the first term of the series whose second and subsequent terms are ,0,3,8 − 0,1 using difference table.
How many terms n be chosen in Maclaurin’s expansion for ex with an error less than
Solve the system of equations:
x + 2y + z = 3
3x − 2y − z4 = −2
2x + 3y − z = −6
using Gauss-Elimination method.
Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:
i) The equation x3 4x 16 = 0 has a root in the interval [3, 4].
ii) The order of convergence of the secant method is 0.62.
iii) For the system of linear equations:
5x + y + 2z = 34
4y − 3z = 12
10x − 2y + z = −4
the matrix is diagonally dominant.
iv) The numerical method:
is relatively stable.
v) The method:
converges to 1.5 for any choice of initial approximation.
Estimate the eigen values of the matrix:
using the Gerschgorin bounds. Also, draw the rough sketch of the region in which the eigen values lie.
Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.