Flashcards
Ch. 6: Group Isomorphisms
Cards in this deck (11)
- 1Define: Group IsomorphismAn isomorphism Phi from G to G\bar is a one-to-one mapping (or function) from G onto G\bar that preserves the group operation.
- 2Four steps to show that a function is Isomorphic1. Define the function from G to G\bar
2. Show that Phi is one-to-one
3.) Show that Phi is onto
4.) Show that Phi is operation preserving - 3Cayley's TheoremEvery group is isomorphic to a group of permutations
- 4Properties of isomorphisms acting on elementsSuppose that Phi is an isomorphism from a group G onto a group G/bar. Then:
Phi carries e to e/bar
Phi carries g^n to Phi(g^n)
a,b in G commute iff Phi(a),Phi(b) commute
G= iff G/bar =
|a| = |Phi(a)| for all a in G
If G is finite, then G and G\bar have the same number of elements - 5Properties of Isomorphisms Acting on Groups1. Phi^(-1) is an isomorphism
2. G is Abelian iff G\bar is Abelian
3. G is cyclic iff G\bar is cyclic
4. If K is a subgroup of G, then Phi(K) = {Phi(k) | k is in K} is a subgroup of G\bar - 6Define: AutomorphismAn isomorphism from a group G onto itself is an automorphism of G.
- 7Definition: Inner Automorphism Induced by aLet G be a group, and let a be an element of G. The function Phi_a defined by
Phi_a(x) = axa^(-1)
for all x in g is called the inner automorphism of G induced by a. - 8Define: Aut(G)The set of automorphisms of a group G
- 9Define: Inn(G)The set of inner automorphisms of a Group G
- 10Are Aut(G) and Inn(G) groups?Yes. Aut(G) and Inn(G) are themselves groups. In fact, Inn(G) is a subgroup of Aut(G)
- 11Under what condition is Aut(Z_n) isomorphic to U(n) ?Whenever n is a positive integer.
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- Mathematics (Abstract Algebra) Flashcards on Ch. 6: Group Isomorphisms, created by William Hartemink on 14/04/2017.
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- William Hartemink
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