Flashcards
§3 Cyclic Groups
Cards in this deck (14)
- 1Definition 3.2: SubgroupG: a group, H⊆G: a subset
H is a subgroup if:
i) H is closed (i.e. a,b∈H then a∙b∈H)
ii) \(1_G\)∈H
iii) For every a∈H, \(a^{-1}\)∈H - 2Definition: Left and right cosetsaH={ah: h∈H}
Ha={ha: h∈H} - 3Proposition 3.1: a~b if aH=bHi) aH=bH ⟺ ah=b for some h∈H
ii) ~ is an equivalence relation
iii) aH= [a] = {\(a^{-1}\)∈G: \(a^{-1}\)~a} - 4Theorem 3.1: LangrangeIf H⊆G, then
\( | H |\) divides \( | G |\) - 5Definition: Index[G:H] = the number of cosets.
- 6Proposition[G:H] =\( | H |\) divides \( | G |\),
if G is finite. - 7Definition 3.5: Group generatorG: a group, \(a_1\),...,\(a_n\) ∈ G.
\(\langle a_1,...,a_n \rangle\) = set of every possible product of powers of \(a_1\),...,\(a_n\). - 8Proposition 3.4:\(\langle a_1,...,a_n \rangle\) is a subgroup
- 9Definition 3.6: Cyclica∈G
\(\langle a \rangle\) is a cyclic subgroup.
If G=\(\langle a \rangle\) for some a, then G is a cyclic group.
⇒ G=\(\langle a \rangle\)={\(a^n\): n∈Z} - 10Theorem 3.3: Cyclic subgroupAny subgroup of a cyclic group is cyclic.
- 11Definition 3.7: Order(1) G: a group
\( | G |\) ("the order of G") = the size of G.
(2) a∈G
\( | a |\) ("the order of a") = |\(\langle a \rangle\)| - 12Definition 3.4: Characterization of cyclic groupsEvery cyclic subgroup is ≅
Z is infinite,
OR
\(Z_n\) if finite. - 13Theorem 3.4: paraphrased from notesa∈G. T.F.A.E:
(1) m = |a|(=|\(\langle a \rangle\)|)\
(2) m is the smallest positive number such that \(a^m\)=e
(3) m is the largest number such that {\(a^0\)=e,\(a^1\),...,\(a^{m-1}\)} are all distinct. - 14Corollary 3.2If \(a^m\)=e, then |a| divides m.
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- Diploma/Certificado de qualificação profissional de nivel 3 Mathematics (Algebra (MATH311)) Flashcards on §3 Cyclic Groups, created by Amy Wogan on 19/03/2016.
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