Examples of familiar, easy-to-visualize manifolds that admit Lie group structures I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a $2$D blanket or a circle/curve or a sphere, a torus etc.
However I have a problem visualizing a Lie group. The best one I thought is $SO(2)$ which as far as I understand just a circle. But a circle apparently lacks distinguished points so I guess there is no way to canonically prescribe a neutral element to turn a circle into a group $SO(2)$.
Examples I saw so far start from a group, describe it as a group of matrices to show that the group is endowed with the structure of a manifold. I would appreciate the other way --- given a manifold show that it is naturally a group. And such a manifold should be easily imaginable.
 A: A good example is the $3$-sphere $\mathbb{S}^3$. In my mind, I think of this first and foremost as a geometric entity, and certainly people were considering spheres before they started to think about groups. As you may know, $\mathbb{S}^3$ can be made into a group by identifying it with the set of unit quaternions
$$\{ q \in \mathbb{H}: \lVert q \rVert = 1\}$$
Then, the group structure on $\mathbb{S}^3$ is inherited from the group structure of the quaternions, making $\mathbb{S}^3$ a Lie group.
A: I find $\mathbb{R} - \lbrace 0 \rbrace$ a helpful example for thinking about Lie groups which aren't connected.
We can see that $(0,\infty)$ is a normal subgroup. We also see that any neighborhood of the identity can generate this subgroup.
A: Some important examples include:


*

*Any finite-dimensional vector space $(\mathbb{V}, +, \cdot)$ over $\mathbb{R}$ or $\mathbb{C}$ is a Lie group under addition $+$; in particular this includes the familiar spaces $\mathbb{R}^n$.

*Any Euclidean space $\mathbb{R}^n$, $n > 1$ admits more than one group structure up to isomorphism. For example, $$(x, y) \ast (x', y') := (x + x',e^{x'} y + y')$$ defines a group structure on $\mathbb{R}^2$, isomorphic to the group $\mathbb{R}^* \rtimes \mathbb{R}$ of affine transformations $t \mapsto a t + b$ with positive slope $a$ under composition. The operation $$(x, y, z) \star (x', y', z') = (x + x', y + y', z + z' + xy')$$ defines a group structure on $\mathbb{R}^3$, and this is usually called the Heisenberg $3$-group. It's easy to verify that both $\ast$ and $\star$ are nonabelian, and so define group structures nonisomorphic to the familiar structures $(\mathbb{R}^n, +)$. In particular, this shows that while we can certainly ask whether a given manifold has a group structure, in general there can be more than one.

*Any torus $\mathbb{S}^1 \times \cdots \times \mathbb{S}^1$ is a Lie group under the direct product multiplication; this is just a quotient of $(\mathbb{R}^n, +)$ by $\mathbb{Z}^n$.

*Isaac has pointed out in another answer (which I include here for some limited semblance of completeness) that $\mathbb{S}^3$ admits a Lie group structure, which we can think of as the group of unit quaternions under quaternionic multiplication. We often call this group $SU(2)$, which we can think of the group preserving a Hermitian form and a (compatible) complex volume form on $\mathbb{R}^4 \cong \mathbb{C}^2 \cong \mathbb{H}$, which we can thus identify explicitly with the group of matrices $A \in GL(2, \mathbb{C})$ satisfying $A^* A = I_2$ and $\det A = 1$.

*The real projective space $\mathbb{RP}^3$ admits a natural group structure, namely that of the orthogonal group $SO(3, \mathbb{R})$. One can see this from the previous example if one identifies $\mathbb{S}^3$ with the double cover $Spin(3, \mathbb{R})$ of $SO(3, \mathbb{R})$.

*By the above example the product $\mathbb{S}^3 \times \mathbb{S}^3$ can be given the group structure $SU(2) \times SU(2)$ (or $Spin(3) \times Spin(3)$), but (in a feature exclusive to this dimension), this group is isomorphic to $Spin(4)$; so, we can regard $SO(4, \mathbb{R})$ (which is double-covered by $Spin(4, \mathbb{R})$) as a quotient of $\mathbb{S}^3 \times \mathbb{S}^3$, namely the one given by identifying a pair $(x, y)$ with its "double antipodes" $(-x, -y)$.

*The punctured spaces $\mathbb{R}^*$, $\mathbb{C}^*$, $\mathbb{H}^*$ are all groups under multiplication; in particular, $\mathbb{C}^*$ is sometimes called $CSO(2, \mathbb{R})$ and is the group of oriented conformal linear transformations of $\mathbb{R}^2$, that is, those preserving angles.

*We can naturally give the interior $\mathbb{D} \times \mathbb{S}^1$ of a solid torus (here $\mathbb{D}$ is the open unit disk) a group structure by identifying a pair $(a, \zeta)$ with the conformal isomorphism $\mathbb{D} \to \mathbb{D}$ defined by
$$z \mapsto \zeta \frac{z - a}{1 - \bar{a} z};$$
these exhaust such isomorphisms, and so this group (under composition) is isomorphic to $PSL(2, \mathbb{R})$.

*Trivially, any group structure on the underlying set of a $0$-manifold is automatically smooth and hence defines a Lie group structure.

*An important nonexample: The sphere $\mathbb{S}^2$ admits no Lie group structure; in fact, the only spheres that admit Lie groups are $\mathbb{S}^n$ with $n = 0, 1, 3$. (The $7$-sphere $\mathbb{S}^7$ comes close---it admits a loop structure with a smooth operation given by octonionic multiplication, but this operation is nonassociative and so cannot define a group structure.)
A: Think of $SO_2$ as the group of $2\times 2$ rotation matrices:
$$ \left[\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right]$$
or the group of complex numbers of unit length $e^{i\theta}$.
You can convince yourself directly from definitions that either of these objects is a group under the appropriate multiplication, and that they are isomorphic to each other.
This group (which is presented in two ways) is a 1-manifold because it admits smooth parametrization in one variable ($\theta$ here).
Why does this group represent the circle $S^1$?  Well, the matrices of the above form are symmetries of circles about the origin in $\mathbb{R}^2$, and the trace of $e^{i\theta}$ is the unit circle in the complex plane.  I think it is best conceptually to think of Lie groups first as groups, and then to develop geometric intuition to "flavor" your algebraic construction.
