Maths WorkshopMaths problems with full solutions

Groups

43 problems · Subgroups, cyclic groups, Lagrange's theorem and group homomorphisms.

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Ex.241Additive and multiplicative notations

#notation #abelian group #convention

Ex.242Checking the group axioms

#group axioms #examples #counterexamples

Ex.475Cyclic groups

#cyclic group #generator #examples

Ex.476Order of an element

#order #cyclic subgroup #element

Ex.477The subgroup criterion

#subgroup #criterion #nonempty

Ex.478Which subsets are subgroups?

#subgroup #examples #counterexamples

Ex.969The center of a group

#center #abelian #subgroup

Ex.970Counting cosets

#cosets #index #counting

Ex.971Groups of prime order

#prime order #cyclic #subgroups

Ex.972The general linear group $GL_n(\mathbb{R})$

#general linear group #matrices #invertibility

Ex.973Group homomorphisms

#homomorphism #kernel #image

Ex.974Homomorphisms between familiar groups

#homomorphism #examples #kernels

Ex.975Subgroups of index 2 are normal

#index #normal #cosets

Ex.976Isomorphism criteria

#isomorphism #bijective homomorphism #criterion

Ex.977Injectivity and kernel

#kernel #injectivity #homomorphism

Ex.978Corollary of Lagrange's theorem

#Lagrange #order of element #corollary

Ex.979Lagrange's theorem

#Lagrange #order #subgroup

Ex.980Order of an element in $\mathbb{Z}/n\mathbb{Z}$

#order #cyclic group #gcd

Ex.981Direct products of groups

#direct product #product group #order

Ex.982Subgroup lattice of $\mathbb{Z}/6\mathbb{Z}$

#subgroup lattice #cyclic group #divisors

Ex.983The symmetric group $S_3$

#symmetric group #permutations #order

Ex.984Transpositions generate $S_n$

#transpositions #generation #symmetric group

Ex.1699Cauchy's theorem applied to $A_4$

#Cauchy #alternating group #elements of order

Ex.1700Cauchy's theorem for groups

#Cauchy theorem #prime order #existence

Ex.1701The Chinese remainder theorem for groups

#Chinese remainder #direct product #cyclic group

Ex.1702Subgroups of a cyclic group

#cyclic group #subgroups #divisors

Ex.1703The dihedral group $D_n$

#dihedral group #symmetries #presentation

Ex.1704The first isomorphism theorem

#first isomorphism theorem #quotient #kernel

Ex.1705The order of $GL_n(\mathbb{F}_p)$

#general linear group #finite field #order

Ex.1706Normal subgroups

#normal subgroup #conjugation #criterion

Ex.1707The normalizer

#normalizer #subgroup #normal

Ex.1708Quotient groups

#quotient group #normal subgroup #cosets

Ex.1709The sign homomorphism

#sign #parity #homomorphism

Ex.1710Simple groups

#simple group #normal subgroup #examples

Ex.1711The special linear group $SL_n(\mathbb{R})$

#special linear group #determinant #normal subgroup

Ex.2147Burnside's lemma

#Burnside lemma #group action #counting

Ex.2148The class equation

#class equation #conjugacy #center

Ex.2149Structure of finite abelian groups

#abelian group #structure theorem #decomposition

Ex.2150Free groups (statement)

#free group #generators #relations

Ex.2151Group actions and orbits

#group action #orbit #stabilizer

Ex.2152Groups of order $p^2$

#p-group #abelian #order

Ex.2153Sylow's first theorem

#Sylow #p-subgroup #existence

Ex.2154Sylow's third theorem

#Sylow #congruence #number of subgroups