+
Related Flashcards
Related Topics
Cards In This Set
| Front | Back |
|
Corollary to Cayley's Theorem
|
Every finite group G of order n is isomorphic to a subgroup of the symmetric group S_n.
|
|
The right coset Ka denotes:
|
Ka = {ka| k in K}, where a is a fixed element of the group.
|
|
Lagrange's Theorem
|
If G is a finite group and K is a subgroup of G, then
|G| = |K| |[G:K]|.
|
|
Def: The index of a subgroup K in a group G
|
Is the number of distinct right cosets of K in G.
|
|
Let p be a prime integer. Then every group of order p is...
|
Cyclic, and therefore isomorphic to Z_p.
|
|
If N is a normal subgroup of a group G, then the following are equivalent:
|
1) Na = aN for every a in G.
2) (a^-1)Na is a subset of N for every a in G.
3) (a^-1)Na is equal to N for every a in G. |
|
G/N denotes...
|
...The set of all right cosets of N in G.
|
|
G/N is a group under the operation (Na)(Nb) = N(ab) if...
|
...N is a normal subgroup of G.
|
|
If G is finite and N is a normal subgroup of G, then the order of G/N is ...
|
...|G|/|N|.
|
|
If G is abelian and N is a normal subgroup of G, then G/N is...
|
...Abelian!
|
|
Let N be a normal subgroup of G. Then G/N is abelian if and only if...
|
...ab(a^-1)(b^-1) is in N for every a, b in G.
|
|
If G is a group such that the quotient group G/Z(G) is cyclic, then...
|
...G is abelian.
|
|
Let f: G->H be a homomorphism of groups with kernel K. Then...
|
...K is a normal subgroup of G.
|
|
Let f: G->H be a homomorphism of groups with kernel K. Then K = <e_G> if and only if...
|
...f is injective.
|
|
First Isomorphism Theorem for Groups:
|
Let f: G->H be a surjective homomorphism of groups with kernel K. Then the quotient group G/K is isomorphic to H.
|
