Can a woman lead davening or bentching (zimmun) if all the people present are her immediate family? Or is it assur? I understand that it would for sure be assur if the people were not her family.
As for the zimmun, there would be three men present.
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There is a discussion in a number of places when women form their own zimmun, if men (less than a zimmun) can remain and answer or not. That is, if there are one or two men present, then there is a question if they can remain and answer the (three or more) women or not. However, if there are three or more men present (as well as three or more women), then the men must form their own zimmun and the women are allowed to form their own zimmun. If there is a minyan of men present, then the women must join the men. There are other commentaries on the matter, but the majority appears to be that an individual woman cannot lead a zimmun unless the men are too ignorant to be able to say birchat hamazon themselves.
The citations below imply that a woman cannot lead a mezuman of men. The individual articles are too long to summarize completely, so I have just included the conclusions. I have also shown the halachos as the Rambam brings them down.
I also saw in the cited articles, that since ten women do not form a minyan, then they cannot add the extra word elokeinu and must join with the ten men, rather than form their own zimmun.
The Mishna Brurah Orach Chaim 199 No. 16 explains that it is a reshus for three women to join together and not a chov. This is also a reason why a woman cannot lead a zimmun of men.
A summary of this can be seen in Women’s Zimmun: An Addendum
What remains clear from all of these sources is that, at the very least, women can form a zimmun; according to many, they are in fact obligated in zimmun. Despite the simple understanding of Berakhot 45b, Halakhah does not permit women to be counted with men to make up the requisite number for a zimmun of three or ten. A woman who eats with three men must remain and share in the zimmun obligation by answering with the men’s zimmun. Women may, however, separate from presence of a men’s zimmun and perform their own, provided that there are fewer than ten men present. Although there are many rishonim who maintain that women have an obligation in zimmun, normative Halakhah has not adopted this view. Nonetheless, zimmun provides an easy opportunity to perform a mitsvah and find greater meaning in Jewish practice.
We can see this from the Rambam who explicitly says.
As an example Rambam Holchos Berachos chapter five
Halacha 1 Women and slaves are obligated to recite grace. There is a doubt whether their obligation stems from the Torah, since [this is a positive mitzvah] that is not linked to a specific time, or whether their obligation does not stem from the Torah. Therefore, they should not fulfill the obligation of grace on behalf of others.
Halacha 7 Women, servants, and children are not included in a zimmun. They may, however, make a zimmun among themselves.
Women’s Zimmun and Whether Men [Who Are Present] Must Leave goes into detail on the subject. This article is too long to give all the details. However, the final summary and conclusion of this article is ...
The result [of all this] is, in my opinion: According to our [the accepted] halakhic ruling that women are only doubtfully [biblically] obligated in birkat ha’mazon, then the stated reason of licentiousness does not refer to the joining together of men and women – unlike the Ritva’s opinion; and since the Sefer Ha’Meorot and the Ohel Moed permit [a man to say the zimmun for women], and we do not find anyone who prohibits this besides the Ritva -- therefore, we can conclude that it is permissible for a man who dined with three women to fulfill their obligation in zimmun for them. The halakhic results, in my opinion, are as follows: (1) One woman and two men, or the opposite [one man and two women] do not join together for zimmun. (2) If three women ate with one or two men, then the women recite the zimmun together and the men may answer, or a man may recite the zimmun to fulfill the women’s obligation. (3) Three women who ate with three men fulfill their obligation through the zimmun of the men, or they may separate and recite a zimmun for themselves provided there are less than ten men. If there are ten men, the men recite the zimmun ba’Shem and the women respond. (4) One woman who ate with three men may not recite the zimmun to fulfill their obligation, but she fulfills her own obligation through the men’s zimmun. (5) Ten women may not recite the zimmun ba’Shem [extended form of zimmun said in the presence of a minyan], in accordance with the ruling of the Rambam and the Shulhan Arukh, and not in accordance with the opinion of the Sefer Ha’Meorot; and also a man who is saying the zimmun for ten women should not recite the zimmun ba’Shem. (6) Women have the option of reciting the zimmun for themselves, and a woman who is not accustomed to doing so is still permitted to say the zimmun; if she followed this custom three times in succession, not counting meals where women were not present [separately] to form a zimmun, then she has established a [permanent] obligation for herself if she did not make a condition that she is saying the zimmun b’li neder ["without a vow" for the future].
Yehuda Herzl Henkin
נָשִׁים מְזַמְּנוֹת לְעַצְמָן, רְשׁוּת. אֲבָל כְּשֶׁאוֹכְלוֹת עִם הָאֲנָשִׁים, חַיָּבוֹת וְיוֹצְאוֹת בַּזִּמּוּן שֶׁלָּנוּ.
ויוצאות בזמון שלנו - ומשמע מדברי היד הקטנה דהיא אסורה לברך ברכת הזימון ולהוציא את האנשים אלא שצריכה לשמוע ולצאת מהזימון שמברכין האנשים:
Women may make zimmun together but when they eat with men, they are obliged to be included in zimmun and do their duty through the zimmun of the men.
and do their duty through the zimmun of the men - and from the Yad Ketana we see that they may not make zimmun and acquit men of their obligation but they must listen and fulfill their duty through listening to the men.
So it seems that a woman cannot lead zimmun for men. Women may make their own zimmun and let the men do their own.