# EXAM 1 MATERIAL

- StudyBlue
- California
- Mills College
- Mathematics
- Mathematics 132
- Berard
- EXAM 1 MATERIAL

**Created:**2017-01-19

**Last Modified:**2017-03-02

- Closure under mult
- Associativity
- Identity Element
- Multiplicative inverses

_{n})

_{n})

^{*}

_{n}(IR)

- 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
^{-1})^{-1}=a

^{n}= e

^{n}≠ e for any positive integer, we say a has infinite order.

- a, b ∈ H ⇒ ab ∈ H
- a ∈ H ⇒ a
^{-1}∈ H

^{n}l n ∈ Z}

^{n}=1.

^{-1}g c

- 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

_{n}

**order**of the permutation θ in Sn is....

^{-1}is in K.

_{H}}

_{(i=1,r)}⊕ R/(d

_{i}) where d

_{1}, ..., d

_{r}are the invariant factors of M, then the p

_{i}

^{αi,j}are the elementary divisors of M.

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