We are asked to show that ______ _____ ________ ______ ______ ___ _________ _____ __________ _____ ______ _______.
___ __________ ___ _________ _________ _____ ________ ______ _____ ____ ____ ____.
______ ________ ______.
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If H and K are normal abelian subgroups of a group, and if H∩ K = {e}, show that HK is abelian. Will the result be still true if we remove the condition that H and K are normal? Justify your answer.
Define an integral domain. Give an example of an integral domain which is not a field.
Define a subgroup of a group. Check whether
is a subgroup of the the group of 2 × 3 matrices over ℂ under addition
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.
पारिवारिक व्यवसाय से आप क्या समझते ह? पारिवारिक व्यवसाय में संघर्ष के कारण बताइए।
एंजेल निवेशकों और वेंचर कैपिटलिस्ट के बीच अंतर बताइए।
(i) Give, with justificaiton, a nilpotent element in R.
(ii) Give, with justification, a zero divisor in 푅 which is not nilpotent.
(iii) What is the order of U(R)?