Give an example, with justification, of a commutative subgroup of a noncommutative group.
In abstract algebra, a subgroup is a subset of a group that satisfies the group axioms under the same operation as the parent group. It is possible for a non-commutative group to have a commutative subgroup.
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Define a relation R on by R =
Check whether R is an equivalence relation or not. If it is, find all the distinct equivalence classes. If R is not an equivalence relation, define an equivalence relation on
Explicitly give the elements and structure of the group
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.
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 .
Give an example, with justification, of a commutative subgroup of a noncommutative group.
Find all the units of
Let Check whether D is a UFD or not.