Let be a finite abelian group and
Prove that
To prove that the set \[ S = \left \{ g\in G \mid (o(g),m) = 1 \right \} \] is a subgroup of \( G \), where \( (G, .) \) is a finite abelian group and \( m \in \mathbb{N} \), we need to verify the subgroup criteria: 1. Identity Element: We must show that the identity element \( e \) of \( G \) is in \( S \). - The order of \( e ______ _____ __________ ____ _______ ____ __________.
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Explicitly give the elements and structure of the group
Let Check whether D is a UFD or not.
Consider the set Define
on
by
i) Check whether is a group or not.
ii) Prove that x ∗ x ∗ x ∗K∗ x (n times) = (1+ x) −1 ∀ n∈N n and x ∈X .
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) where
is the Euler-phi function.
ii) If and
are groups, and
is a group homomorphism, then
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and ,then
v) Every element of has order at most n .
vi) If R is a ring and I is an ideal of R , then I xr = rx ∀ x ∈ I and r ∈ R .
vii) If is a product of an even number of disjoint cycles, then sign
viii) If a ring has a unit, then it has only one unit.
ix) The characteristic of a finite field is zero.
x) The set of discontinuous functions from to
form a ring with respect to pointwise addition and multiplication.
Let G be a group of order 56 . What are all its Sylow p-subgroups? Show that G is not simple, i.e., G must have a proper normal non-trivial subgroup.
Let
i) Show that M is an ideal of R .
ii) Show that if a|5 / or b/|5 , then 5| for ,a . b ∈
iii) Hence show that if N is an ideal of R properly containing M , then N = R .
iv) Show that is a field, and give two distinct non-zero elements of this field.
Let R be a commutative ring with unity and r ∈ R . Prove that using the Fundamental Theorem of Homomorphism. Hence show that