1. “I shall be sanctified “toch” / in the midst of, the children of Israel.” This verse is connected to the verse by Korach which states, “Remove yourselves from “toch” / the midst of, this wicked Eidah / congregation.”
We know that the minimum number for an Eidah / congregation is ten because of the statement by the spies: How long for this evil Eidah / congregation?” (Talmud, Meggilah 23b; Vayikra 22:32, Bamidbar 16:20, Bamidbar 14:27)
2. How do we know that Hashem doesn’t reject communal prayer? It is written, “Behold G-d, Might, He does not reject.” Similarly it is written “You have redeemed me from my foes, because there were many with me.” (Talmud, Brachos 8a, Iyov 36:5, Tehillim 55:19)
3. One may pray in any language that he understands. This is true in communal prayer. But privately, one may not pray in Aramaic. This is because communal prayer goes straight to Hashem, while private prayer goes through the angels, and they don’t like to deal with Aramaic.
(Shulchan Aruch 101:4, with Mishnah Berurah 14,15,18)
4. Rav Yitzchak asked Rav Nachman, Why didn’t you come to synagogue to pray?” Rav Nachman answered, “I was too weak.”
Rav Yitzchak asked, “So why didn’t you gather a minyan where you were?” Rav Nachman answered, “It was too difficult to do so.”
Rav Yitzchak asked, “So why didn’t you ask that someone should notify you when the communal prayers began?” Rav Nachman asked, “What is concerning you?”
Rav Yitzchak explained, “The verse states: May my prayer come before Hashem at a favorable time. Rabban Shimon ben Yochoie taught, ‘When is it a favorable time? When the public is praying.’” (Talmud, Brachos 8a)
5. Reish Lakish said, “Anyone who has a synagogue in town and does not frequent it, is called a bad neighbor.” (Talmud, Brachos 8a)
6. The main application of communal prayer is when the ten people are reciting the Amidah together. Not as people think, that the main goal of communal prayer is to be able to say Kaddish, Kedisha, and Borchu.
(Mishnah Berurah 90:28, see however Ishei Yisroel 12:7 and 8:13, and Teshuvos V’Hanhagos 1:102 teffila bitzibur, teffilas hatzibur)
7. All ten people must be in the same room. (Shulchan Aruch 55:13)
8. If a person is passing by a synagogue and hears kaddish, kedusha, or borchu, he should answer with them, as long as there is nothing dirty that separates them. The requirement to be in the same place is on the original ten men. (Aruch Hashulchan 55:17)
9. A captive who was given authorization to attend services for one day should take the first possible day, and he should not postpone the authorization for Yom Kippur or Purim. (Mishnah Berurah 90:28)
If there is no Minyan
10. I was asked by the members of a small city that can’t get a Minyan every day, but they can gather six or seven people, if there is any point in doing so.
It seems that as long as there are three people, they are within the statement “Hashem does not reject the prayers of the group.”
(Tesuvos V’Hanhagos 2:57)
11. If they began saying Kaddish or Kedusha, and some people walked out, they may complete what they started, as long as most of a minyan remains. In fact, if they began the repetition of the Amidah with ten, they may complete it, recite the kaddish, as well as the kaddish with tiskabeil.
Nevertheless, it is wrong for people to walk out. About them it is written, “Those who forsake Hashem shall perish.”
(Shulchan Aruch 55:1-2, Yeshaya 1:28;
See Shevet Haleivi regarding someone who is trying to help make the Minyan)