Group Theory Midterm Study Guide 3

Here's some examples of different types of groups, properties, and other bits of information that I might need to know for the exam.

Abelian Groups: A group G is Abelian iff ab=ba for all a,b in G.
Examples:

1. Z(n) under regular addition
   
2. U(n) under regular multiplication
3. Aut(Z(n)) under function composition

Properties:

1. Iff G is Abelian, then P(G) is Abelian.
   
2. If G is Abelian, then the subgroup H is necessarily Abelian.
   

Cyclic Groups: A group G is cyclic iff there exists an element a such that 'a'=G.
Examples:

1. In Z(n) with element k, if gcd(n,k)=1 (n,k relatively prime), then k is a generator of Z(n).
   
2. If |G| is prime, then G is cyclic.
   
3. Every subgroup of a cyclic group is cyclic.
   
4. For each positive divisor k of n, the set 'k/n' is the unique subgroup of Z(n) with order k; these are the ONLY subgroups of Z(n).
   
5. S(n) is cyclic.
   

Properties:

1. G is cyclic iff P(G) is cyclic.
2. Moreover, G='a' iff P(G)='P(a)'; that is to say, a generator maps to another generator.
   

Normal Groups: A subgroup H is normal to a group G iff xHx^-1=H, for all x in G.
Examples:

1. Any subgroup H of an Abelian group G is normal to G.
   
2. The centralizer Z(G) is always normal to G.
   
3. A(n) is always normal to S(n).
   
4. R(n) is always normal to D(n).
   
5. SL(2,R) is always normal to GL(2,R).
   

Isomorphisms: An isomorphism P is a mapping from a group G to a group P(G), where P is one-to-one, onto, and operation preserving.
Properties:

1. The identity always maps to the identity.
   
2. G is cyclic iff P(G) is cyclic.
   
3. G is Abelian IFF P(G) is Abelian.
   
4. G and P(G) have the same number of elements of each order, if G is finite.
   
5. G and P(G) have the same order.
   
6. |a|=|P(a)|; that is, P preserves the order of an element.
   
7. For a,b in G, a,b commute iff P(a),P(b) commute.
8. For all a in G, P(a^n)=(P(a))^n.
9. If K is a subgroup of G, then P(K) is a subgroup of P(G).
   
10. Aut(Z(n)) is isomorphic to U(n).
    

Cosets: Let H be a subset of a group G with element a; then, aH={ah| h is in H}. If H is a subgroup of G, then aH is a left coset of G. Similar definitions exists for Ha and aH(a^-1).
Properties:

1. a is in aH.
2. aH=H iff a is in H.
   
3. aH=bH iff a is in bH.
   
4. aH=bH or aH&bH={0}.
   
5. aH=bH iff (a^-1)b is in H.
   
6. |aH|=|bH|.
   
7. aH=Ha iff H=aH(a^-1).
   
8. aH is a subgroup of G iff a is in H.
   
9. The number of distinct cosets (left, right, OR whatever) of H in G is |G|/|H| (Lagrange's Theorem).
   

Lagrange's Theorem and Corollaries:
Lagrange's Theorem: Let G be a group with a subgroup H. Then, |H| divides |G|; also the number of distinct cosets of H in G is |G|/|H|.
Corollary 2: |a| divides |G|.
Corollary 3: All groups of prime order are cyclic.
Corollary 4: a^|G|=e.

External Direct Products: Let G1,..., Gn be a finite collection of groups. The external direct product of G1,..., Gn, written as G1#...#Gn, is the set of all n-tuples for which the ith component is an element of Gi and the operation is componentwise.
What That Means: Think of the Cartesian product. For example, Z(2)#Z(3)={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)}, where (1,1)(1,2)=(0,0), because 1+1mod2=0 and 1+2mod3=0.
Properties:

1. The order of an external direct product is the product of the orders of the individual groups.
   
2. The order of an element of an external direct product is the least common multiple of the orders of the components of the element.
   
3. G#H is cyclic iff |G| and |H| are relatively prime.
   
4. Let m=n1*...*nk. Then, Z(m) is isomorphic to Z(n1)#...#Z(nk) iff ni and nj are relatively prime when i does not equal j. For example, Z(2)#Z(5)#Z(7)==Z(2)#Z(35)==Z(10)#Z(7)==Z(14)#Z(5)

This last property is still a little mysterious to me, but I'm coming around.
I need to go study for Discrete Math, now, but I might post some discrete stuff on here later.
Ciao, folks.
p.s.: I again apologize for the blatant lack of formality in the formatting, Deal with it.