3. Subgroups & Homomorphisms Flashcards
1
Q
Subgroup
A
A subset H of G that forms
a group under the group operation of G.
2
Q
Subgroup Properties
A
- H is a group under the group operation of G
2. H is nonempty and for all x,y in H, we have xy^-1 in H.
3
Q
Subgroup Test
A
Showing the 2 subgroup properties are true.
4
Q
Homomorphism
A
A map between 2 groups that preserves the operations of said groups.
5
Q
Isomorphism
A
A homomorphism that is also a bijection. (1-1 and onto)
6
Q
Image of Homomorphism
A
A subgroup of H and its kernal is a subgroup of G
7
Q
Cayley’s Theorem
A
Every finite group G is isomorphic to a subgroup of the Symmetric group acting on G
8
Q
Injective Group Homomorphism
A
if and only if its kernal is trivial
