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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 .
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.
Find all the units of .
Check whether or not
is a field.
यदि D10 में r एक परावर्तन है, तो D10 में के दो अलग-अलग सहसमुच्चय दीजिए।
Find all the units of .
Is , for any two ideals I and J of a ring R? Give reasons for your answer.
List two distinct cosets of in D10, where r is a reflection in D10.