Karteikarten
§2 Groups
Karten in diesem Stapel (14)
- 1Definition 2.4: Group axiomsG: a set, *: an operations.
G is a group if:
0) G is closed under * (i.e. a*b∈G)
i) * is associative (i.e. a*(b*c)=(a*b)*c)
ii) There is an identity element e∈G
iii) For every a∈G, there is \(a^{-1}\)∈G, such that a*\(a^{-1}\)=\(a^{-1}\)*a=e - 2Proposition 2.1: IdentityThere is only one identity
- 3Proposition 2.2: InverseEvery element of G has only one inverse.
- 4Definition 2.5: AbelianA group \(\langle G,* \rangle\) is abelian if * is commutative.
- 5Definition 2.6 & 2.7\(M_n(R)\) = set of nxn matrices with coefficients in R.
\(GL_n(R)\) = set of nxn invertible matrices with coefficients in R - 6Proposition 2.3∀ a,b∈G, \((a*b)^{-1}\) = \(a^{-1}\)*\(b^{-1}\)
- 7Proposition 2.4For every a∈G, \((a^{-1})^{-1}\)= a
- 8Proposition 2.5, 2.6, and CorollaryLet G be a group and g∈G
i) If a,b∈G, g∙a≠g∙b and a∙g≠b∙g
ii) For any h∈G, gx=h has a solution (i.e. x=g-1h)
iii) G → G, {\(a_1\),...,\(a_n\)}→ {\(ga_1\),...,\(ga_n\)} is one-to-one
Similarly, {\(a_1\),...,\(a_n\)}→ {\(ag_1\),...,\(ag_n\)} is one-to-one - 9Proposition 2.7For any q,r∈Z and a∈G,
\(a^q\)x\(a^r\)\=(a^{q+r}\) - 10Corollary 2.2\((a^n)^{-1}\)=\(a^{-n}\)
- 11Recall: Cartesian Product\(S_1\)x\(S_2\)={(s,t) : s∈\(S_1\) and t∈\(S_2\)
- 12Definition: Direct ProductIf (G,∎) and (H,△) are groups:
Define * on GXH as
(\(g_1\),\(h_1\))*(\(g_2\),\(h_2\)) = (\(g_1\)∎\(g_2\), \(h_1\)△\(h_2\))
* is well-defined,
thus G*H is a group. - 13Theorem: Direct ProductA direct product is a well-defined group.
- 14Definition 2.8: Group IsomorphismsG={e,a,b} and H={E,A,B}. φ:bijection.
Define:
φ(e)=E
φ(a)=A
φ(b)=B
is an isomorphism
when φ satisfies φ(x*y)=φ(x)*φ(y),
φ is a homomorphism.
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- Diploma/Certificado de qualificação profissional de nivel 3 Mathematics (Algebra (MATH311)) Karteikarten am §2 Groups, erstellt von Amy Wogan am 19/03/2016.
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- Amy Wogan
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