Yes and no.

The key word in your question is "public" that changes EVERYTHING. The following paragraphs assume a private company, as in you own 75% and your brother owns the remaining 25%. So a private, not public company.

There's actually 3 "roles" here. First there's your role as a shareholder. This is a person who owns a portion of the company, in this case it's 75% but some shareholders might only own .00001%. Every share you own gets you 1 vote when it comes time to elect board members.

Second there's your role as an employee of the company, normally you'd hire yourself as CEO.

Third there's the board members, and since you are 75% owner you're likely going to elect yourself chairman of the board. It's important to note here, you elect board members based on rules set out in the corporate charter (these are the rules of the company as set out when the company was founded). Every individual board election is voted on by all shareholders. SO a 75% shareholder can control every single board election, therefore 75% of the shares gets you 100% of the board seats.

So this person would be, majority shareholder, chairman of the board and CEO. 3 roles, 1 person. Each role gives you certain powers, but those powers are not unlimited.

Board members then vote on things, such as who to hire as CEO and when to fire the CEO. Since you are CEO, the board members that you elected should vote for you. BUT board members, like any elected official, does not need to vote as they have promised they will vote.

In the event that people you got elected don't vote your way, there would be a process outlined in the corporate charter on how to replace a board member, normally based on the votes of other board members. So if someone didn't vote as you'd like, you'd try to kick them out.

So to answer your question. If the board members you elect turn against you, it's likely that you can replace them but it's going to take some time. So they would have the ability to fire you, but not long term. You'd just start replacing board members who voted against you until you had enough votes to hire yourself back as CEO.

So lets pose a question. Say I own 75% of the shares and my idiot brother owns 25%. My brother is an idiot but people love him and he's extremely popular.

Over time I've elected 7 board members. Myself, as chairman. Both my parents, my 2 cousins, and a husband/wife couple that are an old family friend. SO 7 people, all voted in by me so in theory I control them... in theroy.

My super likeable brother though, he decides he wants to run the company. So he turns on the charm and gets my parents and cousins to agree to vote for him as CEO. A board meeting is called and someone puts forward a motion to fire the CEO, that's ME so I vote no and so do the family friends. But my parents and cousins vote yes. They then vote to hire my brother as CEO. My brother thinks he has won, I am furious.

So I'm no longer CEO, but I'm still board chairman and majority shareholder. Per the company rules board members are elected to 1 year terms and the elections happen at the annual general meeting, but that's not for months.

I have a few options. I could wait the few months, fire my parents and cousins from the board, appoint my wife and 3 children and then fire my CEO brother and hire myself back as CEO. Boom, problem solved but it took a few months.

But my role as chairman likely gives me some powers to address this faster. Most companies would have a process to remove a member of the board. The key here is that since I control the company I also control these rules, so as long as I had a very thoughtful lawyer when the company was founded (lets assume I did) then the chairman can likely remove any board member at any time and then appoint a temporary board member to serve until the AGM where we can properly elect someone.

So assuming I had the foresight to do that, I would use my chairman powers to fire my parents and cousins, appoint my wife and children as temporary board members to replace them, and rehire myself as CEO.

All of this would likely happen in the same meeting where I got fired as CEO.

So to answer your question, yes and no. They can be kicked out, but it's VERY difficult and likely could not happen forever.

But way back at the beginning I noted this would only be for a private company. Once a company wants to offer shares to the general public, and be listed on a stock exchange they need to agree to MANY rules and regulations.

One of those rules speaks to having an "independent" board of directors. This is why many business people, political figures and other "important" people become board members in retirement. It's not a large time commitment, you get paid, and your prior work experience is generally used to frame you as independent. Look at this link https://www.apple.com/ca/leadership/ at the bottom of the page it lists Apple's board of directors.

There's Tim Cook himself, many high level employees of other large companies. There's people from Google, Johnston & Johnston, Northrop Grumman, Boeing, BlackRock (among others). Then there's Al Gore (former VP to Clinton). So public companies "should" have independent board members. Having proper independent board members who have their own reputations outside of their job as board members. SO with board members like that, they might take a moral stand against a CEO even though that CEO is a majority shareholder.

Might, key word there is "might".