EXAM 1 MATERIAL
Last Modified: 2017-03-02
- Closure under mult
- Identity Element
- Multiplicative inverses
- G, H abelian means that GxH is abelian
- If G, H have finite order, GxH = order (G) x order (H)
- G has a unique identity
- Cancellation holds in G
- ab = ac ⇒ b = c
- ba = ca ⇒ b = c
- Each element in G has a unique inverse.
- (ab)-1=(b-1 a-1)
- a, b ∈ H ⇒ ab ∈ H
- a ∈ H ⇒ a-1 ∈ H
- Φ-1 is an iso
- Order of G = Order of H
- If G is abelian, H is abelian
- If G is Cyclic, H is cyclic
- If G has a subgroup of order n, then H has a subgroup of order n
- the order of a in G = the order of Φ(a) in H
- f(eG) = eH
- f(a-1) = f(a)-1
- Im f is a subgroup of H
- if f is injective, then G is isomorphic to Im f
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