A T1 space need not be locally Hausdorff; an example of this is an infinite set given the cofinite topology.
Let X be a set given the particular point topology. Then X is locally Hausdorff at precisely one point. From the last example, it will follow that a set (with more than one point) given the particular point topology is not a topological group. Note that if x is the 'particular point' of X, and y is distinct from x, then any set containing y that doesn't also contain x inherits the discrete topology and is therefore Hausdorff. However, no neighbourhood of y is actually Hausdorff so that the space cannot be locally Hausdorff at y.
If G is a topological group that is locally Hausdorff at x for some point x of G, then G is Hausdorff. This follows from the fact that if y is a point of G, there exists a homeomorphism from G to itself carrying x to y, so G is locally Hausdorff at every point, and is therefore T1 (and T1 topological groups are Hausdorff).
^Niefield, Susan B. (1991), "Weak products over a locally Hausdorff locale", Category theory (Como, 1990), Lecture Notes in Math., 1488, Springer, Berlin, pp. 298–305, doi:10.1007/BFb0084228, MR1173020.