Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example.
(a) Every subgroup of S3 is normal.
(b) Every abelian group is cyclic.
(c) In a ring with unity, the sum of any two units is a unit.
(d) If a field has characteristic p, p a prime, the field is finite.
(e) If every element in group has finite order, the group is finite.
(a) False. Not every subgroup of $S_3$ is normal. Consider the subgroup $H = \{e, (12)\}$. Conjugating by $(13) \in S_3$, we get $(13)(12)(13) = (23) \notin H$. Hence $H$ is _____ ___ ____ _______ _______ __________ _________ _____ ________ ____.
_________ _____ _______ _____ ______ _______ ____ ____ _______ __________ ____.
________ __________ ____ __________ ____ ___ _____ _____ ____ ___ _______ _______.
__________ _____ _________ _______ ___ _________ ______ ______ ______ _______ _______ __________.
________ ______ ________ ___ ____ ____ ____.
__________ _____ _______ _______ _________ _______ _______.
___ _______ _____ _______ __________ __________ _____ _______ ___ _________.
_______ ______ __________ _____ _____ ____ _________.
_____ __________ ___ _____ _______ ___ ______ ______.
____ _________ ________ __________ __________ _______ ___ _________ ________.
___ ________ ____ __________ _____ ____.
_________ ________ _________ ____ _________ ______ _____ __________ _______ _______ __________ __________.
_______ __________ ____ _______ ________ _________ ____ ________.
__________ _________ ___ _____ ______ _____ ________ ________ _______.
Get Full Answer on WhatsApp
Define a semigroup. Give an example of an infinite semigroup.
Find the gcd of the polynomials x4 + 3x + 2 and x³ + 3x² + 5x + 3.
Define an abelian group. Give an example of a non-abelian group. (You don’t need to prove that your example is a group. You have to only prove that it is non-abelian.
Let R = Z20
Find the inverse of the matrix:
using LU decomposition method with u11 = u22 = u33 = 1.
Let S¹ = {z ∈ C | |z| = 1} and Un = {z ∈ C* | z" = 1} for n ∈ N. Check that Un ⊆ S¹. Further, show that Un ≤ S¹.
Let R be a ring (not necessarily commutative) and I and J be ideal of R. Show that I ∩ J and I + J = {a + bla ∈ I, b ∈ J} are ideals of R.