What is the order of
Let's solve these numerically. ### Problem (i) Find the order of \( 14 \) in \( \mathbb{Z}_{24} / \langle \bar{8} \rangle \) #### Step 1: Understand the Structure of the Quotient Group The given group is \( \mathbb{Z}_{24} \) (integers modulo 24) and the subgroup is generated by \( \bar{8} \). This means the subgroup \( \langle \bar{8} \rangle \) consists of multiples of 8: \[ \langle \bar{8} \rangle = _________ ______ ______ ____ ___ ____.
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Show that in a group G of odd order, the equation x2 = e has a unique solution. Further, show that x2 = g has a unique solution ∀g∈G,g ≠e .
Prove that every ideal I of a ring R is the kernel of a ring homomorphism of R.
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)
i) If A and B are two sets such that A ⊆ B, then A × B = B.
ii) If S is the set of people on the rolls of IGNOU in 2016 and T is the set of real numbers lying between 2.5 and 2.55, then T SU is an infinite set.
iii) The set {x
x=1 1(mod 30)is a group with respect to multiplication (mod 30).
iv) If G is a group with an abelian quotient group G/N, then N is abelian.
v) There is a group homomorphism f with Ker f
and Im f
{0}.
vi) There is a 1 – 1 correspondence between the odd permutations of S35 and the even permutations of S35.
vii) If R is a ring such that , a =− a ∀a ∈ R then R is Boolean.
viii) Given any ring R, there is an ideal I of R such that R/1 is commutative.
ix) If S is an ideal of a ring R and f a ring homomorphism from R to a ring R' , then
x) ‘ring’, as we now define it, was first presented to us by Dedekind.
a) The table below is a Cayley table for the group ({e,a, b,c,d},∗). Fill in the blanks.
| * | e | a | b | c | d |
| e | e | - | - | - | - |
| a | - | b | - | - | e |
| b | - | c | e | e | - |
| e | - | d | a | a | b |
| d | - | - | - | - | - |
c) Check if 
is an abelian group with respect to matrix
Can there be a homomorphism from ? Give reasons for your answer
What is the order of
Let F be the ring of all functions from to
w.r.t. pointwise, addition and multiplication. Let S be the set of all differentiable functions in F. Check whether S is
i) a subring of F,
ii) an ideal of F.