Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
Let (G) be a cyclic group generated by (g), i.e., (G = \langle g \rangle), and let (H) be a non-trivial subgroup of (G). Since (G) is cyclic, every element of (H) can be written as a power of (g). Let (h) be a non-identity element ____ ____ __________ _________ ________ _____.
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की अभिसरण कोटि निर्धारित कीजिए।
Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.
Find the inverse of the matrix using Gauss-Jordan method.
Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix using the power method. Take (1, 0, 0)T as the initial approximation and perform 4 iterations.
निम्नलिखित आंकड़ों के लिए, गाउस पश्चांतर विधि का प्रयोग करके को अंतर्वेशी करने वाला बहुपद प्राप्त कीजिए :
| x | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |
| f (x) | 1.40 | 1.56 | 1.76 | 2.00 | 2.28 |
Construct a field with 125 elements.
Which of the following statements are true, and which are false? Give reasons for your answers.
i) If k is a field, then so is .
ii) If R is an integral domain and I is an ideal of R, then .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and is of degree
, then f(x) has exactly n roots in R.