Consider nZ. This set is closed under addition. Moreover, it contains 0 and for each element x also its opposite -x. So, nZ is a subgroup of Z.

The elements of degree 0 form a subgroup of (Q[X],+).

If we consider all elements in Q[X] that take the value 0 at x, then these elements form also a subgroup of the additive group on Q[X].

By the results of Section 5.3 we find A_n to be a subgroup of S_n.

If m is less than n, then we can think of S_m as consisting of those permutations in S_n that fix all x with m < x. So, we can view S_m as a subgroup of S_n.

Similarly we can view A_m as a subgroup of A_n.

Let SL_n(R) denote the set of n by n matrices with real coefficients and determinant 1. Then each element in SL_n(R) has an inverse with respect to matrix multiplication. This inverse has also determinant 1. The product of two elements of SL_n(R) has also determinant 1. Hence SL_n(R) is a subgroup of GL_n(R), called the special linear group.

The subset of matrices of determinant -1 or +1 also forms a subgroup of GL_n(R).

The subset of upper (or lower) triangular matrices of GL_n(R) or SL_n(R) is closed under multiplication and inverses and hence a subgroup of GL_n(R) or SL_n(R), respectively.

Consider a regular n-gon . The rotations over 2k/n, k = 0, ..., n-1, around the center of form a subgroup with n elements of D_n.