सिद्ध कीजिए कि चक्रीय समूह के प्रत्येक अतुच्छ उपसमूह का परिमित सूचकांक होता है। इस तरह सिद्ध कीजिए कि चक्रीय नहीं है
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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.
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 .
Let be a fixed odd permutation in S10. Show that every odd permutation in S10 is a product of
and some permutation in A10.
. 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.
या H अपरिमित है;
निम्नलिखित कथनों में से कौन-से कथन सत्य हैं और कौन-से असत्य हैं? अपने उत्तर की पुष्टि कीजिए।
i) किसी भी वलय R और के लिए,
।
ii) किसी भी वलय में कम-से-कम दो अवयव होते हैं।
iii) यदि R एक तत्समकी वलय है और I, R की एक गुणजावली है, तो R/I का तत्समक वही होता है जो कि R का तत्समक है।
iv) यदि एक वलय समाकारिता है, तो यह (R, +) से (S, +) तक एक समूह समाकारिता है।
v) यदि R एक वलय है, तो से R तक कोई भी वलय समाकारिता आच्छादक होता है।
केली प्रमेय की सहायता से एक ऐसा क्रमचय समूह ज्ञात कीजिए जो कि कोटि 12 वाले चक्रीय समूह के तुल्याकारी हो।