Left translation and conjugation are two actions in which G acts on its own members. In the former, a moves x onto a*x. In the latter, a moves x onto a*x/a.
Now we see the reason for our backwards convention. Let ab act on x by translation. This is a acting on bx, or abx. Similarly, ab conjugates x via (ab)x/(ab), or abx/b/a.
Using translation, any finite group can be represented as a permutation group. The induced homomorphism is 1-1, since only 1 produces the identity permutation.
When the action is conjugation, the stabilizer of x is the centralizer of x, or all elements that commute with x. The conjugacy class of x is the orbit of x. Again, orbits partition the group.
The number of orbits, or the number of conjugacy classes, is the class number of the group.
The "class equation" states that the sum of the sizes of the conjugacy classes is |G|, and each class size divides |G|. The former is a restatement of orbits partitioning the group. The latter comes from |stabilizer|×|orbit| = |G|, as described in the previous section. Note that the orbit has size 1 iff x is in the center of G. We sometimes write the class equation as |G| = |center| plus the sum of the sizes of the nontrivial conjugacy classes.
Since conjugation is an inner automorphism, it maps subgroups to subgroups. Thus a group can act on its own subgroups by conjugation. The stabilizer of a subgroup is its normalizer. This does not mean the group elements are fixed; only that the subgroup is mapped onto itself. In other words, x*H/x = H, which is the definition of a normalizer. Recall that H is normal in its normalizer.
Earlier in group theory, we proved that the group of inner automorphisms is isomorphic to G mod its center. Although it was not specifically stated at the time, this proof uses group action. Each x in G maps to a permutation of the elements of G, that is, the inner automorphism implemented by xG/x. The trivial automorphisms correspond to the center of G, which becomes the kernel of the action. Thus the inner automorphisms become the quotient group G/center(G).