Gemara

We say that the Shechinah dwells among any 10 Yisraelim.

Berachos 21b: Rav Huna holds that an individual (praying with a Tzibur) says Kedushah. R. Yehoshua ben Levi holds that he does not.

(Rav Ada bar Ahavah): An individual does not say Kedushah. It says "v'Nikdashti b'Soch Bnei Yisrael" -- every matter of Kedushah needs 10 men.

(Ravnai): We learn from a Gezeirah Shavah "Toch-Toch". It says about the Meraglim "Hibadlu mi'Toch ha'Edah ha'Zos." Just like there, there were 10 (evil Meraglim), also here there must be 10 (Yisraelim).

Megilah 23b (Mishnah): The following require at least 10 men: to be Pores Al Shma, Yored Lifnei ha'Teivah (the Shali'ach Tzibur repeats Shemoneh Esre aloud), Birkas Kohanim, Kri'as ha'Torah and Haftorah in the Navi.

Rishonim

Rif and Rosh (Megilah 13b and 3:7): The Yerushalmi says that if one began (one of these matters in the Mishnah) with 10 and some left, we finish the matter. It says about this (leaving) "those who abandon Hash-m will perish."

Ran (DH v'Ha): The Mishnah lists separately Pores Al Shma and Yored Lifnei ha'Teivah to teach that they are separate matters. If one began to be Pores Al Shma with 10, and some left, he may finish Pores Al Shma, but he may not be Yored Lifnei ha'Teivah.

Gra (OC 55:3): Everything in the Mishnah is a separate matter. We do not begin any of them if less than 10 remain.

Rosh (ibid): Rashi explains that Pores Al Shma is if 10 came to the Beis ha'Keneses after saying Shema Yisrael. One of them says Kaddish, Borechu, and Birkas Yotzer Ohr. R. Tam says that if six did not hear Kaddish and Borechu, even if the rest already heard, we can say them. He proves this from Maseches Sofrim. It says 'we need 10 for Kaddish and Borechu. Raboseinu in the west (Bavel) say that seven suffice. Some say that six suffice.' Even though it seems that the latter two opinions argue with the first Tana and our Mishnah, and hold that l'Chatchilah six or seven suffice, R. Tam holds that they do not argue. Six or seven amidst the 10 did not yet hear Kaddish and Borechu. In Sefer ha'Yashar, R. Tam says that he found written in Agada that we say Borechu if six did not hear it. Talmidei Rashi say in his name that we are Pores Al Shma even for one. Even if he said Shema, he can be Pores, just like a Shali'ach Tzibur who prayed prays again aloud, even though nowadays we all know how to pray (and we are not Yotzei through the Shali'ach Tzibur). This proof is not so solid, for R. Yochanan taught that it would be ideal if a person prayed the entire day. Also, Chazaras ha'Shatz enables people to answer Kedushah and Modim. Maseches Sofrim also says that if nine or 10 heard Kaddish or Borechu, and after Tefilah one who did not hear it said it among them, and they answered, he was Yotzei. The Ri says that R. Tam did not want to be Pores Al Shma since all heard Kaddish and Borechu except for him. Perhaps Maseches Sofrim discusses b'Di'eved. I say that there is no reason to distinguish like from b'Di'eved. I say that the latter opinions there argue with the first Tana and our Mishnah, and hold that l'Chatchilah six or seven suffice. Why shouldn't nine who heard join with one who did not? He says Kaddish or Borechu, and he is Mekadesh Hash-m amidst 10!

Rambam (Hilchos Tefilah 8:4): Tefilah b'Tzibur (one prays aloud, and the others answer) requires 10 adults, including the Shali'ach Tzibur. We require that the majority did not pray yet. Similarly, we say Kedushah, Kri'as ha'Torah with Berachos before and after, and read the Haftorah from the Navi only with 10.

Rambam (5): Kaddish and Birkas Kohanim are only with 10.

Rambam (6): Regarding all these matters of Kedushah, if they started with 10 and some left, even though it is forbidden, the rest may finish.

Rosh (Berachos 3:18): An individual may say "Kadosh..." in Seder Kedushah (in u'Vo l'Tziyon ). He is like one who reads a verse and translates it. Some say that an individual does not say Kedushah in Birkas Yotzer Ohr. He says only '...Omrim b'Yir'ah Kadosh u'Voruch. Others say that he says it. It seems that this is correct, for it is a mere account of the angels' praise. However, 'Nakdishcha...' (we will sanctify You like the angels do) is only with 10.

Poskim

Shulchan Aruch (OC 55:1): We do not say Kaddish with less than10 adult males. The same applies to Kedushah and Borechu.

Kaf ha'Chayim (7): L'Chatchilah one should get at least six who did not yet hear them. If one cannot, we say them even for one person.

Mishnah Berurah (2): If less than 10 prayed or learned, and now there are 10, one should say at least three verses before saying Kaddish. Pesukei d'Zimrah and the Kaddish after Ma'ariv are exceptions.

Shulchan Aruch (2): If one began Kaddish or Kedushah with 10 and some left, we finish the Kaddish or Kedushah he began if the majority remain.

Beis Yosef (DH u'Mah): The Ran says that presumably, the Yerushalmi permits to finish only if most of the 10 remain.

Beis Yosef (DH veha'Rav): The Avudraham says in the name of Teshuvas ha'Rambam that if they started Birkas Yotzer with 10 and some left, they say Kedushas Yotzer. The same applies to Kedushah in Tefilah.

Mishnah Berurah (7): If one started Yishtabach or Ashrei before Minchah, and some left, the Pri Megadim was unsure whether one may say Kaddish. Several Acharonim forbid.

Rema: It is forbidden to leave if 10 will not remain. "Those who abandon Hash-m will perish." If 10 will remain, he may leave.

Mishnah Berurah (12): It suffices to stay until the matter is finished, e.g. one who joined for Borchu need not wait for Chazaras ha'Shatz.

Mishnah Berurah (14): He may leave only if he heard Kedushah and all the Kaddeshim until Aleinu.

Shulchan Aruch (3): If one began Avos and some left, he says even Kedushah.

Kaf ha'Chayim (35): He does not say Seder Kedushah after Tefilah, like the Rambam (in the Beis Yosef above), unlike the coming Rema.

Kaf ha'Chayim (36): If nine are not listening to Chazaras ha'Shatz, his Berachos are close to Levatalah. We do not say so when some left, for they are unable to listen.

Rema: If some left after he began Yotzer, the Shali'ach Tzibur does not begin Tefilah aloud, for Tefilas Yotzer already finished. (If they left) after he began Tefilah aloud and Kedushah, they may finish all Seder Kedushah and full Kaddish afterwards, for it pertains to Tefilah, for it includes 'accept the prayers.' We do not read the Torah, for this is an independent matter.

Beis Yosef (ibid. and DH v'Chosav): We say the Kaddish after Tefilah only if there are 10, for it is not a part of what they began with 10. The Rashba (1:95) says that if they began Avos with 10, they say Kedushah. Terumas ha'Deshen (15) says that if 10 began Shacharis, and some left after Kedushah, the rest may finish Seder Kedushah and Kaddish, for these are the end of Shemoneh Esre. This is why the Shali'ach Tzibur says full Kaddish only after Seder Kedushah. The Kaddish refers to Shemoneh Esre, for it includes 'Tiskabel' (accept our prayers). However, if they began Kaddish and Borechu and Yotzer with 10, and some left (and similarly at Ma'ariv), we do not Yored Lifnei ha'Teivah and say Kedushah (on Motza'ei Shabbos, Seder Kedushah) and Kaddish. Birkos Kri'as Shema and Tefilah are independent. Similarly, on Monday and Thursday, if they began Tefilah with 10, and some left before Kri'as ha'Torah, we do not read the Torah. Even though full Kaddish is after Kri'as ha'Torah, Kri'as ha'Torah is an independent matter. At Minchah, we read the Torah (on Shabbos or fast days) before Tefilah! It is unlike Seder Kedushah, which is like Tefilah and praise of Hash-m. We can say that it gets the law of Tefilah. There is no proof from the delay to say full Kaddish (that Seder Kedushah is part of Tefilah). When there is Musaf, we say full Kaddish right after Tefilas Shacharis, for in any case we will say full Kaddish after Musaf, and it will refer to Seder Kedushah and Kri'as ha'Torah.

Mishnah Berurah (10): The Rema said 'after he began Tefilah aloud and Kedushah.' It suffices that he began Tefilah aloud. Perhaps he teaches that even if he began Kedushah, the majority of 10 must remain.

Gra (5): The Rema said 'and Kedushah' to teach that Kedushah is a part of Tefilah. It is not independent.

Mishnah Berurah (17): Even saying Kaddish and Borchu is considered starting Yotzer.

Rema: Tefilas Ma'ariv and the Kaddish after it do not pertain to Kri'as Shema and its Berachos.

Taz (3): Therefore, if the began Ma'ariv with 10 and some left, they say the Kaddish before Shemoneh Esre, for it pertains to Kri'as Shema, but not Kaddish after Shemoneh Esre.

Magen Avraham (3): Kaddish before Shemoneh Esre of Ma'ariv was enacted for people who leave. If 10 do not remain after Aleinu, we do not say the Kaddish after it, for it is only a custom, and all the more so Kaddish after Shir ha'Yichud of Mizmor (Shel Yom).

Kaf ha'Chayim (38): We (Sefardim) do not say even the Kaddish before Shemoneh Esre if 10 do not remain.