Review

 A subgroup H of a group G is called .......... if aH=Ha, for all  inG .

• A normal subgroup of G

• Trivial subgroup of G.

• The Centralizer of 𝑎 in 𝐺

• The quotient group of G by H

The subgroup H of G is normal, then the set of left cosets of H in G is itself a group called ……….

• The left coset of H in G containing a

• A normal subgroup of G

• The index of a subgroup H in G

• The quotient group of G by H

• Trivial subgroup of G .

• A normal subgroup of G

• The kernel of a homomorphism 𝜙

• The alternating group

• The left coset of  H in G containing a

• The quotient group of G by H

• The index of a subgroup H in G

• A normal subgroup of G

The group of even permutations is called ……….

• Cyclic group

• The alternating group

• Abelian

• Trivial subgroup of G

• Abelian

• The alternating group

• Trivial subgroup of G .

• Cyclic group

Cayley’s Theorem states “Every group is ………. to a group of permutations”.

• A normal subgroup of G

• Trivial subgroup of G.

• Abelian

• Isomorphic

The number of left cosets of a subgroup in a group is called ……….

• The index of a subgroup H in G

• A normal subgroup of G

• The alternating group

• The quotient group of  by

Let G be a group and let a be any element of G. Then the subgroup <a> is called …………….

• Abelian

• Trivial subgroup of G.

• The Centralizer of 𝑎 in 𝐺

• The cyclic subgroup of G generated by a.

If a group has the property that ab=ba for every pair of elements  and , we say the group is …………..

• The Centralizer of 𝑎 in 𝐺

• The alternating group

• Isomorphic

• Abelian