Requirement of Minyan by Kerias Hatorah:[1]
One may not read from the Torah [with its blessings] with less than ten adult men above the age of Mitzvos.[2]

Paying attention: At least ten men must be paying attention to the reading in order for the Minyan to be valid.

If lost Minyan in middle of reading:[3] If one began the reading with a Minyan of ten men, and in the middle of the reading some of them left, leaving them without a Minyan, the remaining congregation is to complete the reading of the Torah. [This however only applies if at least six people remain.[4] It is considered that one has begun the reading if the words of Baruch Ata Hashem has been said in the first blessing of the first Aliya, however it does not suffice if merely Bachu was recited.[5]] One may continue calling up the remaining Aliyos with the blessings said before and after the reading.[6] [Thus, on Shabbos, one is to call all seven Aliyos. It is however forbidden to add more Aliyos than necessary in such a situation.[7] Accordingly, one is not to call an 8^th Aliya for Maftir on Shabbos, and rather the seven Aliyos should include the Maftir.[8] The Haftorah is to be read without a blessing before or afterwards.[9] After the Torah reading, one may say half Kaddish.[10]]

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[1] Michaber 143/1

[2] Michaber ibid; Rambam Tefila 2; Ketzos Hashulchan 26/14

[3] Michaber ibid

[4] Peri Chadash 143; P”M 143 A”A 1; Ketzos Hashulchan ibid; See Admur 55/3

[5] P”M 143 A”A 1; Derech Hachaim; Ketzos Hashulchan ibid

[6] M”A 143/1; Kesef Mishneh ibid; See Machatzis Hashekel ibid

[7] M”A ibid; Ketzos Hashulchan ibid

[8] M”A ibid as explained in Machatzis Hashekel ibid and Levushei Serud ibid

[9] Elya Raba 143/2; P”M 143 A”A 1; Machatzis Hashekel 143/1

The reason: As the Haftorah is a Mitzvah in its own right and is not dependent on the reading of the Torah. [ibid]

Opinion of M”A: It is possible to learn in the M”A ibid that he holds one is to also read the Haftorah with its blessings. It is however also possible to explain his intent is that it should be read without its belssings. [P”M and Machatzis Hashekel ibid]

[10] Shaareiy Efraim; M”B 143/2; Ketzos Hashulchan ibid

[11] Chayeh Adam 31/11 questions this matter; Aruch Hashulchan 69/14; Binyan Shlomo 35; Igros Moshe 1/38

[12] The reason: As the obligation of Kerias Hatorah is on the public and not on a private individual. Hence, only if a full Minyan is obligated in the Mitzvah, because they did not yet hear Keria, can a Keria be performed. [Chayeh Adam ibid; Aruch Hashulchan ibid; Poskim ibid]

[13] Kaf Hachaim 143/3 in name of Lev Chaim 2/25

[14] The reason: As one requires Ruba Deminkar. [ibid]

[15] Biur Halacha 143/1 “Bepachus” based on Ran; Ashel Avraham 69; Maharsham 1/175; 2/97; Lev Chaim 2/25; Levushei Mordechaiy 1/39; Ketzos Hashulchan 26/14; Keren Ledavid 16; Har Tzevi 1/52; Shaareiy Teshuvah 566/4 in name of Machazik Bracha regarding Taanis Tzibur  that 6-7 people fasting suffices; See Yabia Omer 8/14; Mishneh Halachos 3/17

[16] The reason: As whenever majority of a Minyan is obligated in a matter we allow the recital of a Davar Shebekedusha. [See Admur 55/3; 69/4-5]

[17] Peri Hasadeh 2/97; Shaareiy Rachamim 7/18; Toras Yekusiel 51; Daas Sofer 1/8; Daas Torah 69 although concludes with Tzaruch Iyun; Mahariy Asad 51; So rule regarding a fast: Bach in name of Aguda; Meiri Megillah 2a; Orchos Chaim [Lunil]; Leket Yosher 114 in name of Terumos Hadeshen; Aruch Hashulchan 566/7; Mahrsham 2/97; Daas Torah 69; Sdei Chemed 1 Klalei Haposkim 13/7; Sefer Haminhagim [English] p. 26; Piskei Dinim 566; Shut Tzemach Tzedek Shaar Hamiluim 8, brought in Shaar Hakolel 9/17; Igros Kodesh 16/313; Luach Kolel Chabad; Hiskashrus 1024 footnote 16

[18] Implication of Maharil end of Shabbos that he once missed Kerias Hatorah and read it to himself after the Minyan. See Daas Torah ibid

[19] See Noda Beyehuda Tinyana O.C. 15; Birkeiy Yosef, brought in Daas Torah ibid