Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic
Cayley's theorem is a fundamental result in group theory that states that every group can be represented as a subgroup of a symmetric group. In this case, we want to find the permutation group to which a cyclic group of order 12 is isomorphic.
A cyclic group of order 12, denoted as C₁₂, is generated by a single _____ ______ _______ ________ __________ _____ _______ ___ ___ _______ __________ _________.
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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 k k.
ii) If R is an integral domain and I is an ideal of R, then Char (R) = Char (R I).
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 f(x) R[x] is of degree
, then f(x) has exactly n roots in R.
Let G be an infinite group such that for any non-trivial subgroup H of G, |G : H | . Then prove that
i) H G
H = {e} or H is infinite;
ii) If g G, g
e, then o(g) is infinite.
For an ideal I of a commutative ring R, define
for some n
}. Show that
i) is an ideal of R
ii)
iii) in some cases.
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 G = .
ii) If x and y are elements of a non-abelian group (G, ) such that x
y = y
x, then x = e or y = e, where e is the identity of G with respect to
.
iii) There exists a unique non-abelian group of prime order.
iv) If (a, b) A
A, where A is a group, then o((a, b)) = o(a)o(b).
v) If H and K are normal subgroups of a group G, then hk = kh h
H, k
K.
Find all the units of
Let 
be a fixed odd permutation in S10. Show that every odd permutation in S10 is a product of and some permutation in A10 .