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Section Results

This page contains theorems and propositions used throughout the course.

Subsection Results

Proposition 12 Basic Group Properties

Proposition 13 Socks-Shoes Principle

Theorem 17 Two-Step Subgroup Test

Theorem 18 One-Step Subgroup Test

Theorem 21 Center is a Subgroup

Proposition 26 \(K_4\) is the Smallest Non-Cyclic Group

Theorem 27 Equality of Powers of Group Elements

Theorem 28 Fundamental Theorem of Cyclic Groups

Theorem 31 Subgroups Generated by Powers of an Element

Theorem 38 Every Permutation is a Product of Disjoint Cycles

Theorem 39 Disjoint Cycles Commute

Theorem 40 Order of a Permutation

Theorem 41 Every Permutation is a Product of Transpositions

Proposition 42 Canonical Generators of \(S_n\)

Theorem 43 Always Even or Always Odd

Theorem 53 Cayley’s Theorem

Theorem 54 Isomorphisms Preserve Element Properties

Theorem 55 Isomorphisms Preserve Group Properties

Proposition 60 Boring But Useful Coset Properties

Theorem 62 Lagrange’s Theorem

Theorem 77 Order of \(HK\)

Theorem 78 Groups of Order 2p are Cyclic or Dihedral

Theorem 79 Criterion for \(G\times H\) Cyclic

Theorem 84 Order of an Element in a Direct Product

Theorem 88 Normal Subgroup Test

Theorem 99 G/Z Theorem

Theorem 100 Properties of Homomorphisms: Elements

Theorem 101 Properties of Homomorphisms: Subgroups

Theorem 103 First Isomorphism Theorem

Theorem 104 Correspondence Theorem - Fourth Isomorphism Theorem

Theorem 105 Diamond Theorem - Second Isomorphism Theorem

Theorem 106 Fraction Theorem - Third Isomorphism Theorem

Theorem 107 Fundamental Theorem of Finite Abelian Groups