Abstract Algebra: Groups and Subgroups

Suppose we have a set of 2 by 2 matrices whose determinants are non-zero. Consider the subset of these matrices whose determinant is 1. Using the two-step subgroup test, verify if this subset is closed under multiplication and if an element belongs to this subset, confirm that its inverse also belongs to this subset.

Given a finite subset of a group, verify if the subset is closed under the group operation to determine if it is a subgroup.

Suppose we have a non-empty subset of a group $G$. Prove that this subset is a subgroup if and only if for all $x, y \in H$, $xy^{-1} \in H$.

Show that the centralizer of $H$, defined as $C_H = \{ g \in G \mid gh = hg, \forall h \in H \}$, is a subgroup of $G$.

Show that the conjugate subgroup $G^{-1}HG$, defined as consisting of elements of the form $G^{-1}hG$ for $h \in H$, is a subgroup.

Show that the inverse of an inverse is itself.

Every element of a group has exactly one inverse.

If a group G has 7 elements, what are the possible orders for its subgroups?