Construct a field with 125 elements.
A field with 125 elements must have order (125 = 5^3), since the number of elements in a finite field is always a power of a prime. Here, (p = 5) and (n = 3). To construct the field ( \mathbb{F}_{5^3} ), start with the prime field ( \mathbb{F}_5 = {0,1,2,3,4} ). Then, ________ ______ _________ _____ _______.
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Give the smallest for which An is non-abelian. Justify your answer.
Let G be an infinite group such that for any non-trivial subgroup H of . Then prove that
i) or H is infinite;
ii) If , then o(g) is infinite.
The velocity of a vehicle beginning from rest is given in the following table for part of the first four. Using Simpson's rule, find the distance travelled by the vehicle in this hour:
|
| 10 | 20 | 30 | 40 | 50 | 60 |
|
| 80 | 60 | 70 | 75 | 70 | 80 |
यदि , तो C1 और C2।
Evaluate by using trapezoidal rule with
and
. Use Romber's m
Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then .
ii) If x and y are elements of a non-abelian group (G, ) such that , then
or
, where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If , where A is a group, then
.
v) If H and K are normal subgroups of a group G, then .
Which of the following statements are true, and which are false? Give reasons for your answers.
i) For any ring R and .
ii) Every ring has at least two elements.
iii) If R is a ring with identity and I is an ideal of R, then the identity of R/I is the same as the identity of R.
iv) If is a ring homomorphism, then it is a group homomorphism from (R, +) to (S, +).
v) If R is a ring, then any ring homomorphism from into R is surjective.