Question
Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:
Let H and K be subgroups of a group G and H' and K' be normal subgroups of H and K, respectively. Then
i)
ii)
iii)
The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.
PART-B (MM: 30 Marks)
(Based on Block 3.)
Answer :
Word Count : 1448
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Related Questions
Construct a field with 125 elements.
क) क्रमविनिमेय वलय R की गुणजावली I के लिए,, किसी
के लिए परिभाषित कीजिए। दिखाइए कि
यहाँ चित्र में दिए गए टेक्स्ट का लिखित रूप है:
i) की एक गुणजावली है।
ii) .
iii) कुछ स्थितियों में
।
ख) क्या वलय R की कोई भी दो गुणजावलियों I और J के लिए,
? अपने
उत्तर की पुष्टि कीजिए।
(5)
यदि D10 में r एक परावर्तन है, तो D10 में के दो अलग-अलग सहसमुच्चय दीजिए।
Is , for any two ideals I and J of a ring R? Give reasons for your answer. Let S be a set, R a ring and f be a 1-1 mapping of S onto R. Define + and
on S by:
.
Show that is a ring isomorphic to R.
Let be a fixed odd permutation in S10. Show that every odd permutation in S10 is a product of
and some permutation in A10.
केली प्रमेय की सहायता से एक ऐसा क्रमचय समूह ज्ञात कीजिए जो कि कोटि 12 वाले चक्रीय समूह के तुल्याकारी हो।
Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
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