group-like elements in the Hopf algebra of the homology of $H$-spaces Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as
$$
S=\{a\in H_*(X)\mid \psi a=a\otimes a, a\neq 0\}.
$$
( I searched online that such kind of elements are called "group-like".)
Question: Is $S$ a basis for $H_0(X)$?
Remark: I meet the question from
The Homology of $E_\infty$-spaces, J.P. May, page 19:
 A: Yes, and in fact you do not need $X$ to be an H-space: the coproduct $\psi$ of $H_*(X)$ exists anyway (if the ring of coefficients is a field), and the definition of a group-like element does not involve the product. So take $X$ to be any space, and define $S \subset H_*(X)$ as in your question (the set of classes in $H_*(X)$ satisfying $\psi(a) = a \otimes a$).
First, a grouplike element is necessarily of degree zero. WLOG one can assume that $a$ is homogeneous (otherwise write $a = a_0 + a_1 + \dots$, where $a_i$ is homogeneous of degree $i$, decompose $a \otimes a$ similarly, and it follows that each nonzero $a_i$ will be grouplike). Then $|\psi(a)| = |a|$, while $|a \otimes a| = 2 |a|$, and $|a| = 2|a| \implies |a| = 0$.
A basis of $H_0(X)$ is given by connected components of $X$ (a standard fact about singular homology). Let $[C] \in H_0(X)$, then by definition of the coproduct $\psi$, one has $\psi[C] = [C] \otimes [C]$ (the Eilenberg–Zilber map is very simple in degree zero), so $[C]$ is grouplike. Reciprocally, suppose that $a \in H_0(X)$. Then $a = \sum \lambda_i [C_i]$, where the $[C_i]$ are classes of connected components and $\lambda_i$ are nonzero scalars. Suppose that $a$ is grouplike, then
$$\begin{align}
\psi(a) = a \otimes a & \implies \sum_{i = 1}^n \lambda_i [C_i] \otimes [C_i] = \left(\sum_{i=1}^n \lambda_i [C_i]\right) \otimes \left(\sum_{i=1}^n \lambda_i [C_i]\right) \\
& \implies \sum_{i = 1}^n \lambda_i [C_i] \otimes [C_i] = \sum_{i=1}^n  \sum_{j=1}^n \lambda_i \lambda_j [C_i] \otimes [C_j].
\end{align}$$
By comparing coefficients, and using that elements of the form $[C] \otimes [D]$ (where $[C]$, $[D]$ are classes of connected components) make a basis of $H_0(X) \otimes H_0(X)$, we find that $n=1$ and $\lambda_1^2 = \lambda_1 \implies \lambda_1 = 1$. Thus $a = [C_1]$ is the class of a connected component.
Conclusion: Grouplike elements of $H_*(X)$ are precisely the classes of connected components of $X$, which are a basis for $H_0(X)$.
