I download Electrum AppImage but it do nothing.
I do not insist. I just like to use it

Can you please explain this? And what type of Linux Mint you are using, default Ubuntu based or Debian based?
You don't need to install anything additional with .appimage files, just click on them, they are similar to portable files in wind0ws OS.
Another option is to check in Linux Mint package manager and search if Electrum wallet is listed there.
I also found few years old topic with Electrum wallet issues on Linux Mint forum:
https://forums.linuxmint.com/viewtopic.php?t=3463301 Hi Is to reply to ABCbits.
2 XFCE Linux Mint 22.1 'Xia' standard installed fresh from USB.
3. I click and it do nothing.
4. package manager giving my "Electrum-GRS"
https://www.groestlcoin.org/groestlcoin-electrum-wallet/ - not trust for my, as i like to install normal Electrum .
5. problem is not same as i can veryficy signature. 23 10 2024, 08:13:40 BST RSA 637DB1E23370F84AFF88CCE03152347D07DA627C