Answer

I’ll tell you my personal view (many halakhic decisors disagree, but in my opinion this is conservatism with no halakhic basis whatsoever).
There is no halakhic impediment to reading from the Torah and receiving an aliyah even within a quorum of men. There is no impediment to all those receiving aliyot being women. This is an explicit rule in the Mishnah (“all count toward the number of seven”), and there is no need for hair-splitting arguments in order to permit it. True, they speak about preventing women from reading because of the dignity of the congregation, but in my opinion that argument is not relevant today (no man is offended by a woman being more learned than he is, since that is extremely common today. And even if someone is offended—that is entirely his problem. If anything, my sense is that the congregation is harmed by excluding women from something there is no reason to exclude them from).
She recites the standard blessings, because there is no difference between her and a man. And in any case, the Ashkenazic custom is that women recite blessings even over commandments they are not obligated in. Especially since they are obligated in the blessings over Torah study (see Shulchan Arukh, Orach Chayim 47).
If you want to look into it more deeply, I’m attaching a very moderate article (I am more radical than it is) that discusses the matter in detail:
 
You can find many more articles on Google (though one should know that not everything written in an article is necessarily correct, of course).
 
As for a women’s quorum, this is discussed in Aryeh Frimer’s article, and here there is a summary, including sources as you requested (taken from here):
 
A different question is whether women can themselves constitute a quorum parallel to a quorum of men, that is, a prayer service including the recitation of “Barekhu,” “Kaddish,” “Kedushah,” and the other matters that require a quorum. In the article “The Status of Women in Jewish Law – Women and a Quorum,” Rabbi Aryeh Avraham Frimer brings three approaches regarding women joining a quorum: [21]
According to the first approach, women join a quorum only in matters for which the exact same obligation applies to them as to men, whether the obligation is Torah-level or rabbinic, as long as the obligation is equal for all. According to this approach, women join a quorum of men because their obligation is identical.
Meiri says: “Anything that requires ten—some say that since women’s obligation in it is equal to men’s, they join the ten… but this does not appear correct,” meaning that some hold this, although it is not Meiri’s own view.[22] In his comments Meiri relies on Rabbi Chaim Eliezer ben Yitzhak, author of Or Zaru’a, who said: [23]
And although regarding zimmun they said [Berakhot 45a], “women recite zimmun for themselves”… because of licentiousness… but for the Megillah they join, though this does not appear correct. It is a simple custom from the words of the Geonim that when reading the Megillah they spread it out like a letter in order to publicize the miracle, and not roll it like a Torah scroll.[24]
[Since the Megillah is opened like a letter in order to publicize the miracle, unlike Torah reading, reading the Megillah is not identical to Torah reading. Therefore there are views that permit women to join in publicizing the miracle, and in the case of the Megillah women can therefore be active participants.]
That is to say, the commandment of reading the Megillah is clearly considered a commandment whose essence is publicity, and therefore all the laws that apply to Torah reading do not apply to it.
Although women are obligated in individual prayer, they are exempt from communal prayer, and therefore according to this definition, which makes their joining depend on publicity, they do not join a quorum for prayer. Accordingly, their prayer as a group cannot include matters of sanctity and will always remain in the category of individual prayer. Regarding Torah reading, according to most halakhic decisors women are exempt, but Rabbi Yosef Teomim, author of Pri Megadim, holds that women do not join a quorum for Torah reading because they are exempt from the reading, unlike the reading of the Megillah, where they are obligated and therefore do join. [25]
The second approach rejects women joining for anything connected to a quorum. The source for this opinion is in the Talmud, tractate Berakhot 45b: “A hundred women are like two men” [that is, a hundred women are considered like two men]. In that passage the Talmud is discussing the obligation of zimmun, and later writes that women recite zimmun for themselves and slaves recite zimmun for themselves once there are at least three people, because there are three “minds,” and three minds are more significant than two men. But Tosafot explains that any number of women will never help for matters that require a quorum, and even a hundred women will still always be considered only as two.
There are additional sources that strengthen this view, such as the Vilna Gaon’s commentary.[26] The claim is that since women are not fit to complete a quorum, they cannot join Torah reading even in cases in which they too are obligated in it, such as the section “Zakhor.” According to this approach, the determination is not made based on the level of women’s obligation, but on the determination that they cannot join a quorum for the sanctification of God’s name.
According to Rabbi Yosef Engel: “The concept of a community depends on inheriting the Land and holding it, for only through that do people connect and gather together, but women have no holding, since they did not receive a portion in the Land, and therefore are not called a community.”[27]
Another opinion within this approach is that of Rabbi Gedaliah Felder, who holds that for a person to be considered part of the community, he must be wholly given over to the public at that moment, and since women are always under the authority of their husbands, they cannot devote themselves to the public and therefore do not join a quorum. [28] In order not to distinguish among women, the same law would apply even to unmarried women.
According to this approach, women cannot conduct any ceremony that requires a quorum, because from the outset they are not permitted to read from the Torah. Therefore even in cases where women are obligated to hear the Torah reading, they cannot join a quorum because they are not fit to complete a holy congregation. Here joining does not depend on obligation, but on their fixed status as women.
The third approach distinguishes between a quorum for “fulfillment” and a quorum for “publicity.” In a regular quorum, a minimum of ten men is required in order to fulfill certain commandments, but there are cases where the purpose of the commandment is only publicity, and in such cases women join a quorum of men even if their level of obligation is not identical to that of men (unlike the first approach).
It seems that this distinction between these two types of quorum was first made regarding the reading of the Megillah, and this is the view of Ran:[29]
And Nachmanides wrote… that all those listed there [in Mishnah Berakhot 23b] are communal obligations, and they are not done unless there are ten, or most of them are obligated in the matter, such as those who did not hear Kaddish and Barekhu. But for Megillah, ten are not required except because of publicizing the miracle… [Unlike other texts that require a quorum for their fulfillment, for the Megillah the quorum is required for publicity, and therefore women certainly participate as well in publicizing the miracle.]
Ritva also holds that women join a quorum for the purpose of Megillah reading: “Therefore here [to join them] to ten is only for the publicizing of the miracle alone… they do join.”[30]
In practice, the great later authorities ruled that women join a quorum for the sake of publicity, such as Megillah reading. A dispute emerges regarding women joining for a matter of sanctity, since this is a classic example of something whose essence is the sanctification of God’s name in public, that is, publicity. If so, do women join for matters of sanctifying God’s name or not? Some associate them with this approach, such as Rabbi Meiselman, but in Rabbi Frimer’s opinion, women joining for the sanctification of God’s name is possible only according to the first approach.[31]
 
Notes:
[19] A.A. Frimer, “The Status of Women in Jewish Law – Women and a Quorum,” Or HaMizrach, vol. 34 [issues 1–2 (120–121)], New York, 1986.
[20] Based on Orach Chayim, siman 55, law 1.
[21] Above, note 19.
[22] Based on Orach Chayim, siman 55, law 1.
[23] Brought by Rema at the end of siman 690.
[24] Rav Hai Gaon, cited there.
[25] Rosh Yosef, Megillah 23a, s.v. “Lema R. Akiva,” Lvov, 18th century, cited in Frimer (above, note 19).
[26] Commentary of the Vilna Gaon to Orach Chayim 199, siman 100, law 6, and see further sources in note 73 of Frimer’s article there.
[27] In Gilyonei HaShas to Berakhot 45a, cited in Frimer, ibid.
[28] Pri Yeshurun, on Tanna Devei Eliyahu Rabbah part 1, on Shema and its blessings, section 15, p. 368, cited in Frimer, ibid.
[29] Milhamot Hashem to Megillah 5a.
[30] Ritva’s novellae to Megillah 4a, s.v. “She’af.”
M. Meiselman, Tradition 52 (3), Fall 1975 [31] cited in Frimer (above, note 19), note 20.
[32] See Frimer (above, note 19), note 71.
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Questioner:
Hello Rabbi Michi,
Following up on my questions about Torah reading for women, I read Rabbi Frimer’s article (and also saw that you posted my question and your answer on your site). It seems to me that Rabbi Frimer is more comfortable with women receiving aliyot specifically within a quorum of men, and not even when all those receiving aliyot are women.
I want to sharpen our questions, because our minyan is about to decide how we are going to conduct ourselves. We are speaking mainly about Simchat Torah, when the Torah reading (in most synagogues I know) is split into several stations according to the number of Torah scrolls, in order to allow as many people as possible (including children) to receive an aliyah.
In the framework of the prayer service we held over the last two years, we split up so that there was one Torah scroll in the men’s section and two scrolls in the women’s section (several hundred Orthodox women came in order to receive aliyot). In the men’s section, only men read and received aliyot, and in the women’s section only women read and received aliyot.
The questions that are important for us to receive a clear ruling on are:
 
1) When there are only women in the space (that is, there is no quorum of men, or there are no men at all), is it permitted for women to read from the Torah and receive aliyot, including saying “Barekhu”? Is saying “Barekhu” one of the matters of sanctity that requires a quorum of ten men? Or can ten women also constitute a quorum?
 
2) According to those who hold that specifically a quorum of ten men is required, does the mere fact that there are men in the men’s section constitute a “sufficient” quorum so that women can say “Barekhu” when receiving an aliyah, or must one make sure that there are ten men specifically intending to answer “Barekhu” at each one of the stations?
 
With blessings and many thanks,
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Rabbi:
But I already answered you according to my opinion. I also added that, as you know, there are other opinions. I think there is a misunderstanding in what you are saying about what a halakhic ruling is. If you expect a description of the views, I already said there are opinions in both directions. Within that, my opinion is one more opinion, exactly like Frimer’s opinion. If for some reason you specifically want my opinion (even though I am not your community rabbi, and of course you have no obligation to accept it), then I wrote it. Now you have to decide what to do. What does it mean to give a clear halakhic ruling? Do you expect me to decide for you whether to follow my opinion or other opinions? But even regarding that decision itself, you could accept or reject my opinion, and we are back to the same point again.
 
I’ll repeat again: in my opinion, a women’s quorum is a quorum for all purposes, and one may read, receive aliyot, and recite blessings in such a quorum. More than that, in my opinion this is much better than a situation in which ten male potted plants are sitting in the men’s section, unrelated to the matter, and thanks to them, so to speak, you get to receive aliyot in the women’s section somewhere else. In such a situation, would the Holy One, blessed be He, dwell in the men’s section while you read and bless Him in the women’s section?! The quorum is supposed to be ten people who are partners in the act of reading and who answer the blessings of the person receiving the aliyah—not clever gimmicks of this kind.
 
Much success,
Michi
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A.:
In honor of Rabbi Michael Abraham
      Following what you wrote regarding aliyot for women, please see the attached article:
https://drive.google.com/open?id=0BwJAdMjYRm7IQmM1N2JSYkRlejg
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Rabbi:
Thank you for the article (which was already familiar to me, at least in part, and I’ve commented on it quite a bit).
I’ve now gone through it again quickly, section by section, and saw that it is good. Although of course there is a great deal to discuss in the usual way of Torah study, I will note here only a few comments (and again, I hope I’ll be forgiven if I err, since I went through it very quickly):
1. It seems to me that at least in some cases, the halakhic decisors who hold that women are exempt from Torah reading do not say this prior to the consideration of the dignity of the congregation, but on the basis of that very consideration. If so, then according to their view the conclusion could come out differently (for if the consideration of the dignity of the congregation falls away—and of course that too requires discussion—then the obligation itself returns). That is how I wondered about the words of the Mishnah Berurah, who brings the Magen Avraham and brings no dissenters, yet still notes that nowadays many did not follow him and women also had the custom of attending. This implies there is no principled dispute here, only a lack of insistence on implementing their obligation for some secondary reason (dignity of the congregation??). Still, this requires further study: why attend if they are not obligated? Is there any prohibition in the matter?! I haven’t checked now.
Be that as it may, regarding receiving an aliyah and even reciting the blessing, there is much room to permit it even if they themselves are not obligated in the reading, as the article says.
2. The Rashba’s statement that you brought—that in a public setting anyone can read, even someone not obligated—should be expanded further. By way of introduction: in commandments imposed on the community, women too are obligated as part of the community, even if these are commandments from which they would otherwise have been exempt (for example, time-bound commandments, as with Hakhel, as implied by Sefer HaChinukh and others). Therefore perhaps with Torah reading too, even if one understands it as part of the law of Torah study (which itself is highly doubtful in my view for several reasons), since it is communal Torah study, women are included in it, and it comes out that even according to the strict law they really are obligated in it, and not merely able to receive an aliyah and recite a blessing.
3. I would also note that regarding the dignity of the congregation, if I understood correctly (I read very quickly), the article mainly discusses two claims: 1. whether it can be waived; 2. whether human dignity overrides it (Sperber’s argument). But a third argument can also be raised: 3. nowadays, in our current situation, there is no issue of dignity of the congregation at all, because the congregation’s dignity is not harmed when women read or receive aliyot. Perhaps the opposite: when they do not receive aliyot, that is what harms the congregation’s dignity. That of course depends on the question of what the dignity of the congregation means, and this is not the place to elaborate.
4. I’ll add further that even if the medieval authorities and the halakhic decisors explain the term “dignity of the congregation” in a certain way, in the end the law still depends on whether in practice the congregation’s dignity is actually harmed or not. For example, even if someone explains that the congregation’s dignity is harmed if someone not obligated receives an aliyah, and even if we assume that women really are not obligated in the matter, still one must discuss what the law is if, as a matter of fact, the congregation’s dignity is not harmed by this in practice (or at least that is how it is perceived today). Is that exactly like a congregation that waives its dignity, such that it depends on whether it can waive it or not (as discussed in the article)? Or perhaps the idea that it cannot waive it is said only when in most cases the congregation does in fact view this as a harm to its dignity; and then there are opinions that if a specific congregation wants to waive it, it cannot. But here, where reality has changed and in all congregations (at least the modern ones) this is not seen as a harm, then according to all views there is no problem of the dignity of the congregation, and it does not depend on whether a congregation can waive it or not. In other words: the question is whether the halakhic decisors’ determination that this is not the dignity of the congregation is factual or normative.

There is also a related question here.