Explicitly give the elements and structure of the group
let's consider the group \( S_n \) (the symmetric group on \( n \) elements) and its normal subgroup \( A_n \) (the alternating group on \( n \) elements), where \( n \geq 5 \).
The quotient group \( S_n / A_n \) consists of cosets of \( A_n \) in \( S_n \). Let's denote the coset \( gA_n \) as _____ ___ _______ ______ ______ ____.
_________ ___ ________ _____ __________ ______ ________ _____ ______ _________ _______ ____.
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Explicitly give the elements and structure of the group
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 be a finite abelian group and
Prove that
Check whether or not A is a subgroup of
i) ii)
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
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.
Give an example, with justification, of a commutative subgroup of a noncommutative group.