Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.
A cyclic group of order 12, denoted (C_{12}), consists of 12 elements generated by a single element (g) such that (g^{12} = e), where (e) is the identity element. Cayley’s theorem states that every group (G) of order (n) is isomorphic to a subgroup of the symmetric group (S_n), the group of _______ __________ ______ ________ ______ _______.
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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.
List two distinct cosets of in D10, where r is a reflection in D10.
Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
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 .
. निम्नलिखित कथनों में से कौन-से कथन सत्य हैं? अपने उत्तरों की पुष्टि कीजिए।
i) यदि एक समूह G अपने किसी उचित उपसमूह के तुल्याकारी है, तो ।
ii) यदि x और y एक अना-बेली समूह (G, ) के ऐसे अवयव हैं जिनके लिए , तो
या
, जहाँ e, के सापेक्ष G का तत्समक है।
iii) अभाज्य कोटि वाला एक और केवल एक अनाबेली समूह है।
iv) यदि , जहाँ A एक समूह है, तो
।
v) यदि H और K समूह G के प्रसामान्य उपसमूह हैं, तो ।
सिद्ध कीजिए कि चक्रीय समूह के प्रत्येक अतुच्छ उपसमूह का परिमित सूचकांक होता है। इस तरह सिद्ध कीजिए कि चक्रीय नहीं है
Let be a fixed odd permutation in S10. Show that every odd permutation in S10 is a product of
and some permutation in A10.