2

This is more to check an argument Since $F_2$ is a group generated by words of two generators, call them $\{a,b\}$ now every other $F_n$ (provided $n\geq 2$) will be all free words of more generators $\{a,b,c,d,...\}$ so it will surely contain words made up of only $a$ and $b$ and attaching one word with only letters $a$ and $b$ to another words containing $a$ and $b$ will still be a word containing only $a$ and $b$, this $F_2$ is closed under group multiplication. Is this argument correct?

El Ruño
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  • Yes! Although if you felt it necessary to check closure under multiplication, it's equally important to check the existence of inverses. – Greg Martin Jan 09 '23 at 23:29
  • well if inverses exist in $\mathbb{F_2}$ assuming it is a group and all elements in of $\mathbb{F}_2$ exist in $\mathbb{F}_n$ then there are for sure already there (that was at least my thinking) – El Ruño Jan 09 '23 at 23:31
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    Yes that's obvious. But what's surprising is that $F_2$ contains $F_n$ as a subgroup, $n\gt2. $ This isn't true for free abelian groups. – calc ll Jan 10 '23 at 00:30
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    Incidentally, fairly famously $F_2$ also contains a subgroup isomorphic to $F_n$ for any $n.$ – Thomas Andrews Jan 10 '23 at 01:04
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    (We usually write $F_n$ for the free group on $n$ generators, and use $\mathbb F_n$ for the field with $n$ elements, if it exists.) – Thomas Andrews Jan 10 '23 at 01:05
  • More interesting is the statement where $\mathbb F_n$ has an isomorphic image on $\mathbb F_2$ – janmarqz Jan 10 '23 at 13:35
  • @eggnog Really?! (I am stunned :D) $F_2$ contains (let's say ) $F_5$ as a subgroup. If I understood you correctly. But how is a word (for example of $F_5$) $aabcde$ containd in the set of all words of only the letters $(a,b)$? – El Ruño Jan 11 '23 at 00:01
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    It's rather counterintuitive. But, if I'm not mistaken, you can use the $a^kba^{-k}$ as a free basis. Also btw, $F_2$ has an infinitely generated subgroup. – calc ll Jan 11 '23 at 00:47
  • take a hike https://math.stackexchange.com/questions/306116/the-free-group-f-2-contains-f-k well not too long hike :D – janmarqz Jan 12 '23 at 11:44

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