Simple facts about generators of groups

Question

There are a couple of simple facts about generators of groups which seem intuitively plausible but which I'm not sure how to establish rigorously. (They emerged from this question.)

If a group $G$ cannot be generated by $n$ elements, how do I show that $G\times H$ also cannot be generated by $n$ elements? And if a group $G$ is generated by $n$ elements, how do I show that its quotient is also generated by $n$ elements?

score 5 · Accepted Answer · answered Jun 16 '18 at 20:34

5

If $G \times H$ is generated by $n$ elements $\{(g_1, h_1), \cdots, (g_n, h_n)\}$, then $g_1, \cdots, g_n$ generate $G$.

If $G$ is generated by $g_1, \cdots, g_n$, then for any normal subgroup $N$ of $G$, $G/N$ is generated by $g_1 N, \cdots, g_n N$.

answered Jun 16 '18 at 20:34

Kenny Lau

25,049

1

The first fact is a particular case of the second one. – YCor Jun 16 '18 at 22:51
@YCor Why? [Additional symbols since Comments must be at least 15 characters in length.] – user557 Jun 16 '18 at 23:26
@user437307 $G = (G \times H) / H$ – Kenny Lau Jun 16 '18 at 23:26

Simple facts about generators of groups

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