# Groups generated by elements - enumerating elements

From Abstract Algebra in GAP by Alexander Hulpke, section 3.23 "Groups generated by elements":

We have seen so far two ways of specifying subgroups: By listing explicitly all elements, or by specifying a deﬁning property of the elements. In some cases neither variant is satisfactory to specify certain subgroups, and it is preferrable to specify a subgroup by generating elements.

Deﬁnition: Let $G$ be a group and $a_1, a_2, ..., a_n ∈ G$. The set

$<a_1,a_2,\ldots,a_n> = \\ \bigg \{ b_1^{\epsilon_1} \cdot b_2^{\epsilon_2} \ldots \ldots b_k^{\epsilon_k} | k \in N_0 = \{0,1,2,3,\ldots\}, b_i \in \{a_1, \ldots, a_n \}, \epsilon_i \in \{ 1, -1 \} \bigg \}$

(with the convention that for $k = 0$ the product over the empty list is $e$) is called the subgroup of $G$ generated by $a_1,...,a_n$.

My question is around the interpretation of the set notation above.

Let's say we have <a,b>. If I'd like to enumerate the set according to the definition above, is this the correct interpretation?

k = 0    ()

k = 1

a
a^-1
b
b^-1

k = 2

a    a
a    a^-1
a^-1 a
a^-1 a^-1

a    b
a    b^-1
a^-1 b
a^-1 b^-1

b    a
b    a^-1
b^-1 a
b^-1 a^-1

b    b
b    b^-1
b^-1 b
b^-1 b^-1

k = 3

...

• yes, that seems to be the correct interpretation, although its hard to tell only with $k=0,1,2$. How do you generate it for larger $k$? – Jorge Fernández Hidalgo May 16 '18 at 3:15
• @dharmatech: You have the right idea. For general $k$, we want all possible products of $k$ terms, where each term is either $a_i$ or $a_i^{-1}$, for some $i$. Hence in your example with two generators $a,b$, each of the $k$ factors must be one of $a,a^{-1},b,b^{-1}$. – quasi May 16 '18 at 3:19
• After you ask a question here, if you get an acceptable answer, you should "accept" the answer by clicking the check mark $\checkmark$ next to it. This scores points for you and for the person who answered your question. You can find out more about accepting answers here: How do I accept an answer?, Why should we accept answers?, What should I do if someone answers my question?. – Shaun Jun 18 '18 at 0:49