Tag Archives: basic course

Given groups G and H, recall we have the product P = G × H and projection maps π1:P → G and π2:P → H. There’s nothing mysterious about the projection maps: these just take (g, h) to g and h respectively. But … Continue reading →

We will talk more about the properties of nilpotent groups, before narrowing down to the concrete examples (as well as non-examples). The first thing we shall prove is: All p-groups are nilpotent. Proof. Let G be a p-group; its centre Z(G) is … Continue reading →

Related to solvable groups is the concept of nilpotent groups. Let G be a group. We say G is nilpotent if there is a sequence of decreasing normal subgroups of G: such that G/Gn+1 commutes with Gn/Gn+1 for all n. Once again, … Continue reading →

First, let us talk about some examples / non-examples of solvable groups. All abelian groups are solvable, since G(1) is trivial. Any group G of order pq for primes p < q is solvable; indeed the Sylow q-subgroup N is … Continue reading →

This topic is rather detached from the rest, but it is sufficiently important that we can’t afford to drop it. Recall, that the finite simple groups form the building blocks of all finite groups. ( Of course we need to solve … Continue reading →

We already know it’s easy to construct an infinite group where every element has finite order. For example, a product of infinite copies of C2. But that’s because we cheat by constructing infinitely many copies of finite-order objects. So let’s constrain … Continue reading →

Warning: this section has almost no proofs! The presentation of a group allows us to describe a group rather economically. For example, only 2 symbols and 3 relations are required to describe the nonabelian group of order 21. Thus there’s … Continue reading →

Let X be a set and F(X) be the free group on X. If Y is  subset of F(X), we consider the normal subgroup N(Y) generated by Y. [ We’ve not mentioned this before, but the intersection of a collection … Continue reading →

Consider a general group G with subset S. Without knowing anything else, what can we say about the generated subgroup ? If S has only one element, we simply get a cyclic subgroup. If S = {a, b}, things start … Continue reading →